Thermomechanical Behavior of the Solidifying Shell and ...ccc.illinois.edu/s/2007_Presentations/13__HIBBELER_Lance... · Thermomechanical Behavior of the Solidifying Shell and Ideal

University of Illinois at Urbana-Champaign • Metals Processing Simulation Lab • Lance C. Hibbeler 1

Thermomechanical Behavior of the Solidifying Shell and Ideal Taper in a

Funnel Mold

CCC Annual Meeting

12 June 2007

Lance C. Hibbeler (MSME Candidate)


Acknowledgements

• Continuous Casting Consortium Members (Nucor, Postech, LWB Refractories, Algoma, Corus, Labein, Mittal Riverdale, Baosteel, Steel Dynamics)

• Begoña Santillana, Arnoud Kamperman, and ArieHamoen (Corus RD&T, Corus Strip Products)

• Dr. Seid Koric• National Science Foundation

– DMI 05-28668

• National Center for Supercomputing Applications (NCSA) at UIUC

• Dassault Simulia, Inc. (ABAQUS parent company)


Outline

• Problem identification

• Simulation overview

• Model parameters

• Analysis procedures

• Model validation

• Discussion of results


Problem Identification

• Approximately 60% of breakouts on the Corus thin slab caster (IJmuiden, The Netherlands) over the past two years have been due to longitudinal face cracks – Mostly occurring in the transition region of the

funnel mold


Longitudinal Face Crack


Breakout from LFC


LFC Mechanisms

• The main mechanism of longitudinal face cracks is non-uniform heat transfer, caused by level fluctuations and/or by several mold phenomena

Shell buckling (excessive taper)

Necking (shell sliding, sticking)


Funnel Mold Terminology

Inner Flat

Inner Curve

Outer Curve

Outer Flat

IF

IC

OC

OF

Top View


Mold Funnel Shapes

0

50

100

0 50 100 150 200 250 300 350 400 450 500

A-Configuration Mold

0

50

100

0 50 100 150 200 250 300 350 400 450 500

B-Configuration Mold

Axes show distance from cold face and distance from centerline in mm.

Thick line is top edge, thin line is bottom edge


LFC Locations

0

1

2

3

4

5

6

7

8

9

0 100 200 300 400 500 600 700

Distance from Center (mm)

Fre

qu

en

cy

(--

)

Inner Flat Inner Curve Outer Curve Outer Flat

January 2005 - January 2007

A-Configuration Molds

Data provided by A. Kamperman of Corus


Project Overview

• The purpose of this work is to use a numerical model to gain insight into the thermomechanical behavior of the solidifying steel shell in a thin-slab funnel mold

• One-way coupled thermomechanical model– Solve heat transfer problem

– Import results, then solve stress problem


Method of Analysis

• Finite element simulation with ABAQUS 6.6– Specializes in nonlinear problems

– Jobs easily scalable with multiple processors

– Robust contact algorithms

– Functionality extended through user subroutines

– Built-in pre-processing capabilities

– Customizable post-processing (with Python)


Thermal Analysis

• Uses isotropic, temperature- and phase-dependent thermal properties– Thermal conductivity

– Specific heat capacity

• Constant density

• Analysis includes rate-of-change effects of temperature-dependent properties

22 2

2 2

T T k T T k Tk k k

x x x x x x T x

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞ ⎛ ⎞= + = +⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠( )p

Tc k T

tρ ∂ = ∇⋅ ⋅∇

∂


Thermal Analysis

• Using simple heat flux profile– Numerical stability

• Fixed mesh of standard four-node heat transfer elements– Currently ~14k elements and ~15k nodes

• Implemented via default ABAQUS heat transfer routines

• Requires ~5 hours (2 CPUs)


Mechanical Analysis

• Uses isotropic, temperature- and phase-dependent mechanical properties– Coefficient of thermal expansion

– Elastic modulus

• Constant density

• Uses Kozlowski III and modified power law constitutive models

• Liquid and mushy zone treated as very low elastic modulus, perfectly plastic solid

• Implemented via Koric’s UMAT subroutine


• Kozlowski III model for austenite [Kozlowski, 1991]

• Modified power law for delta-ferrite [Parkman, 2000]

Constitutive Equations

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

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

( )

32 1 4

1

31

32

33

24 4 5

1/ sec. % exp 4.465 10

130.5 5.128 10

0.6289 1.114 10

8.132 1.54 10

(% ) 4.655 10 7.14 10 % 1.2 10 %

oo

f T Kf T Ko o o

o o

o o

o o

f C MPa f T K K T K

f T K T K

f T K T K

f T K T K

f C C C

ε σ ε ε −

−

−

−

⎡ ⎤= − − ×⎢ ⎥⎣ ⎦

= − ×

= − + ×

= − ×

= × + × + ×

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

( )( )

2

5.52

5.56 104

5

4

1/ sec. 0.1 (% ) 300 (1 1000 )

% 1.3678 10 %

9.4156 10 0.3495

1 1.617 10 0.06166

no m

o

o

MPa f C T K

f C C

m T K

n T K

ε σ ε−

−

− ×

−

−

= +

= ×

= − × +

= × −


Mechanical Contact

• Uses the ABAQUS surface-to-surface contact algorithm with a “softened” exponentialpressure-overclosure relationship– Mold is the master surface

– Steel shell is the slave surface (softer)

• Tangential friction factor* of µfrict = 0.16

• Uses a vector-based algorithm, so the curved surface of the funnel is easy to handle

* [Meng, Thomas, et. al., CMQ 45-1 p 79-94]


Mechanical Analysis

• Uses the exact same mesh as the thermal analysis, using the hybrid formulation (more stable in the liquid region) of four-node generalized plane strain elements

• Requires ~35 hours (2 CPUs)

• Working to optimize UMAT subroutine, though it is already many times faster than using the ABAQUS-native creep algorithm


Finite-Element Mesh

CornerNear inside

of funnel

Domain discretization error ~0.05%


Dimensionality

• Full three-dimensional simulation is a very large and complicated task

• Domain can be two-dimensionalized by relating time to depth (x-y-z to x-y-t):

( )shell men cz t z v t= + ⋅

( ) ( )sin2osc

mold men c oscS

z t z v t tω= + ⋅ + ⋅

• Commonly called “traveling slice analysis”

• Simulated mold includes oscillation effects:2osc oscfω π= ⋅ ⋅


Traveling Slice – Half Speed


Dimensionality

• Some losses in simplifying from 3-D to 2-D– Out-of-plane bending

– Not a problem – investigating longitudinal cracks

What actual steel wants to do What the simulated steel does

[Koric, 2006]


Process Parameters

• Casting speed 5.5 m/min• Pour temperature 1545.0 ºC• Oscillation stroke 7.2 mm• Oscillation frequency 331 cycles/min• Strand width 1200 mm• Narrow face taper 1.0 %/m• Meniscus depth 104.2 mm• Time in mold 10.86 s


Meniscus Depth

0

20

40

60

80

100

120

140

3.75 4 4.25 4.5 4.75 5 5.25 5.5 5.75

Casting Speed (m/min)

Dis

tan

ce

Be

low

Mo

ld T

op

(m

m)

Minimum = 90 mmMaximum = 125 mmAverage = 104.2 mmStd. Dev = 7.79 mm

Measurements from molds with 0-5 mm of wear

Data provided by A. Kamperman of Corus


Steel Properties

• Properties for Corus grade F12L from CON1D/CON2D models– Carbon content 0.045 %wt

– Solidus temperature 1507.51 ºC

– Liquidus temperature 1530.45 ºC

– Solidus/liquidus based on modified Clyne-Kurzsimple segregation model [Won and Thomas, 2001]


Steel Properties

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

Percent Weight Carbon

Tem

per

atu

re (

ºC)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

600 700 800 900 1000 1100 1200 1300 1400 1500 1600

Temperature (ºC)

Ph

ase

Fra

ctio

n (

--)

fa fd

fg fl

[Won and Thomas, MTB, 2001]


Steel Properties

0

200

400

600

800

1000

1200

1400

1600

600 700 800 900 1000 1100 1200 1300 1400 1500 1600

Temperature (ºC)

Sp

ecif

ic H

eat

(J/k

g·K

)

28

30

32

34

36

38

40

42

600 700 800 900 1000 1100 1200 1300 1400 1500 1600

Temperature (ºC)

Th

erm

al C

on

du

ctiv

ity

(W/m

·K)

in liquid259.35

Lf = 247 kJ/kg

[Harste, 1989] [Harste, 1989]


Steel Properties

0

4

8

12

16

20

24

28

600 700 800 900 1000 1100 1200 1300 1400 1500 1600

Temperature (ºC)

Co

effi

cien

t o

f T

her

mal

Exp

ansi

on

(10

-6/º

C)

Tref = 20 ºC

0

25

50

75

100

125

150

600 700 800 900 1000 1100 1200 1300 1400 1500 1600

Temperature (ºC)

Ela

stic

Mo

du

lus

(GP

a)

[Harste, 1989], [Harste and Jablonka, 1988], [Jimbo and Cramb, 1993]

[Mizukami, 1977]


Boundary Conditions

• Two-fold symmetry• Heat flux imposed on the shell surface

• Ferrostatic pressure applied on the shell surface, pulling the shell towards the mold

• Mold hot face converted to a time-dependent displacement function imposed on the two-dimensional domain

• All other faces stress free and/or insulated


Two-Fold Symmetry

x-symmetry plane y-symmetry plane

x

y


0

1

2

3

4

5

6

7

8

9

0 100 200 300 400 500 600 700 800 900 1000

Distance Below Meniscus (mm)

He

at

Flu

x (

MW

/m2 )

0.0 1.1 2.2 3.3 4.4 5.5 6.6 7.7 8.8 9.9 11.0

Time in Simulation (s)

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

Qavg = 2.72 MW/m2

Heat Flux Profile


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


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

Simple Profile

2.72 MW/m2

[Santillana, et. al., AISTech 2007]


Corner Treatment

• Uniform heat flux around the shell– Cools off corners excessively

• Solution: linearly decrease heat flux to 50% of nominal value at the corner over 20 mm on both faces

100%

100% 50%

20 mm


0

10

20

30

40

50

60

70

80

0 100 200 300 400 500 600 700 800 900 1000


Pre

ss

ure

(k

Pa

)

0.0 1.1 2.2 3.3 4.4 5.5 6.6 7.7 8.8 9.9 11.0


Ferrostatic Pressure

Delayed ~1s to allow outermost row of elements to solidify


0

2

4

6

8

10

12

14

0 100 200 300 400 500 600 700 800 900 1000


Dis

pla

ce

me

nt

(mm

)

0.0 1.1 2.2 3.3 4.4 5.5 6.6 7.7 8.8 9.9 11.0


Wide Face

Narrow Face

A-Configuration Molds

Mold Hot Face Displacement


Model Validation

• Compare numerical model to known analytical solution to verify process– Elastoplastic stress model [Boley and Weiner, 1963]

• One-dimensional domain, not including a mold, constrained against bending

• Pure material poured with no superheat into an isothermal mold– Yield stress linearly decreases with temperature

to zero at melting point


One-Dimensional Domain

Prescribed heat flux Three sides insulated

Zero perpendicular displacement

Stress-free edge

Nodes constrained to equal displacement

Domain

Molten steelMold wall


1D Test Case Properties

MPa20.0Yield stress at mold temperature

ºC1495.0Initial temperature

ºC1000.0Mold temperature

GPa40.0Elastic modulus at mold temperature

UnitsValueProperty/Condition

kg/m37500.0Density

--0.3Poisson’s ratio

m/(m·ºC)20.0E-6Coefficient of thermal expansion

W/(m·K)33.0Thermal conductivity

kJ/kg272.0Latent heat

J/(kg·K)661.0Specific heat

Tsol = 1494.38 ºC

Tliq= 1494.48 ºC

h = 220 kW/(m2·K)

T∞= 1000.0 ºC


1D Test Case Properties

0

10

20

30

40

50

1000 1100 1200 1300 1400 1500

Temperature (ºC)

Ela

sti

c M

od

ulu

s (

GP

a)

14 GPa above 1494.38 ºC

40 GPa below 1494.38 ºC

0

5

10

15

20

25

1000 1100 1200 1300 1400 1500

Temperature (ºC)

Yie

ld S

tre

ss

(M

Pa

)

20 MPa at 1000 ºC

35 kPa at 1494.38 ºC


FEA vs. Analytical 1D Results

1000

1100

1200

1300

1400

1500

0 5 10 15 20 25 30Distance Below Shell Surface (mm)

Te

mp

era

ture

(ºC

)

ABAQUS 5s ABAQUS 10s ABAQUS 15s ABAQUS 20s ABAQUS 25s ABAQUS 30s

Analytical 5s Analytical 10s Analytical 15s Analytical 20s Analytical 25s Analytical 30s


FEA vs. Analytical 1D Results

-20

-15

-10

-5

0

5

10

15

0 5 10 15 20 25 30Distance Below Shell Surface (mm)

Str

es

s (

MP

a)

ABAQUS 5s ABAQUS 10s ABAQUS 15s ABAQUS 20s ABAQUS 25s ABAQUS 30s

Analytical 5s Analytical 10s Analytical 15s Analytical 20s Analytical 25s Analytical 30s


Realistic 1D Analysis

• Allow temperature-dependent properties and time-varying boundary conditions for a more realistic simulation, using a finer mesh than what could be used practically (at this point in time) in two dimensions– 300 x 1 row of elements over 30 x 0.1 mm domain

– Typical results away from funnel features

• Very fine mesh provides the ability to determine if an effect is a numerical error


Realistic 1D Results

1000

1100

1200

1300

1400

1500

1600

0 3 6 9 12 15

Distance Below Shell Surface (mm)

Te

mp

era

ture

(ºC

)

2.5s5s7.5s10.86s

γ

δ+γ

LL+δ

δ

-12

-9

-6

-3

0

3

6

0 3 6 9 12 15

Distance Below Shell Surface (mm)S

tre

ss

(M

Pa

)

2.5s5s7.5s10.86s


Realistic 1D Results

-12

-9

-6

-3

0

3

6

0 2 4 6 8 10 12 14 16 18 20 22 24

Distance Below Shell Surface (mm)

Str

es

s (

MP

a)

10.86s

LL+δγ

δ+γ

δ

So

lidu

s


Temperature Results

t = 10.8s (mold exit)

A-mold

B-mold

Parallel mold


Temperature Results

t = 10.8s

(mold exit)

Solidus

9.6 mm


Temperature Results

t = 10.8s (mold exit)

A-mold

Slight two-dimensional heat transfer in funnel transition region


1000

1050

1100

1150

1200

1250

1300

0 50 100 150 200 250 300 350 400 450 500 550 600 650Distance from Center of Funnel (mm)

Te

mp

era

ture

(ºC

)

A-mold, 5s

Inner Flat Outer Curve Outer FlatInner Curve

Surface Temperature

Slight two-dimensional effects


Comparison of Temperature Variations Across Each Funnel

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 100 200 300 400 500 600 700 800 900 1000


Ma

x A

bs

olu

te D

ev

iati

on

fro

m P

ara

llel

Mo

ld S

he

ll S

urf

ac

e T

em

pe

ratu

re (

ºC)

0 1.1 2.2 3.3 4.4 5.5 6.6 7.7 8.8 9.9 11

Time in Mold (s)

A-mold B-mold

Wiggles are numerical effect

Effect grows with time, despite decreasing heat flux and increasing funnel radius


Funnel Stress Development

Tension

Compression

Compression

Tension

From Bending/Unbending


Funnel Shape DevelopmentFrom Change in Length

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0 100 200 300 400 500 600 700 800 900 1000


Str

ain

(m

m/m

)

A-Mold B-Mold

Per

cent

cha

nge

in m

old

perim

eter

in

funn

el tr

ansi

tion

regi

on


Stress Transformation

• Unless otherwise noted, stress/strain results presented on graphs are transformed to a plane tangent to the funnel surface via:

( ) ( )cos 2 sin 22 2

x y x yx xy

σ σ σ σσ θ τ θ

+ −′ = + +

σx

σy

τxy

σx’

σy’

τx’y’

x’

xθ


WF Surface Stress Profiles

6.00

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

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

Distance from Centerline (mm)

Ta

ng

en

tia

l Str

es

s (

MP

a)

A-mold B-mold

Time = s


WF Strain/Displacement

10.85

-5.0

-4.5

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

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

Distance from Centerline (mm)

x-D

isp

lac

em

en

t (m

m)

-20

-18

-16

-14

-12

-10

-8

-6

-4

-2

0

Ta

ng

en

tia

l Str

ain

(m

m/m

)

A-Mold, ux B-Mold, ux

A-Mold, e B-Mold, e

Time = s


Conclusions

• Larger funnel radii provides more uniform heat transfer in the funnel region

• The present antisymmetric funnel design concentrates strains in the funnel region

• Narrow face taper is equal in importance to funnel shape


Upcoming Work

• Apply a more realistic, experimentally validated heat flux profile from CON1D [Santillana, Thomas, Hamoen, Hibbeler, Kamperman, van der Knoop, AISTech 2007]

• Include the thermally distorted mold shape

• Fully coupled thermo-mechanical model

• Study of mechanical effect of oscillations

Documents

Thermomechanical Behavior of the Solidifying Shell and ...ccc.illinois.edu/s/2007_Presentations/13__HIBBELER_Lance... · Thermomechanical Behavior of the Solidifying Shell and Ideal