ISIJ International, Vol. 39 (1999), No. 5, pp, 445-454

AFinite Element Model for 2-Dimensional Slice of Cast Strand

HeungNamHAN.Jung-Eui LEE.1) Tae-jung YE0,2) YoungMokWON.2)Kyung-hyun.KIM 3) Kyu

HwanOH2)and Jong-KyuYOON2)

Sheet Products &Process Research Team,Technical Research Labs., POSCO,Pohang-shi. Kyungbuk, 790-785, Korea.1) Iron &Steelmaking Research Team. KwangyangResearch Labs,, POSCO.Kwangyang-shi, Cheonnam,545-090. Korea.2) School of Materials Science and Engineering and Research Institute of AdvancedMaterials, Seoul National University.Kwanak-ku, Seoul 151 -742, Korea. 3) Technology DevelopmentGroup, M/UFAB. Department, Semiconductor R&D.SamsungElectronics, Yongin, Kyungi-Do 449-900, Korea.

(Received on July 27. 1998; accepted in fl77al form on February 2. 1999)

A two-dimensional thermo-elasto-plastic finite element model for the analysis of thermo-mechanicalbehavior of strand in continuous casting process has been developed. The model incorporates the effect ofmicrosegregation of solute elements on hot tears using a thermo-mechanicai model of mushyzone and ~and y phases. A finite element technique for the liquid region of the slice of strand, which can take theferrostatic pressure due to gravity force into account, wasproposed, The model successfully analyzed thethermo-mechanical behavio! of the solidifying shell of siab in the moldduring the solidification, Thecalculatedresults, such as the deformed geometries, the temperature history, the stress distribution and the formationof air gap between solidifying shell and mold in the continuous casting process of slab, were obtained.Thesewere comparedwith the reported observations,

KEYWORDS:finite element material model for liquid and mushy zone; ferrostatic pressure; micro-segregation; deformation of solidifying shell.

l .Introduction

The continuous casting process has been mathemati-cally modeledto increase the understanding of the roles

of important variables in the process, improve the designof continuous castlng system, and eliminate casting de-fect. Especially, manythermo-mechanical finite elementanalyses had beendeveloped to understand the deforma-tion behaviors of the solidifying shell in the continuouscasting process. I ~ 11)

Tszeng and Kobayashi4) developed a finite elementmodel to compute the thermo-mechanical state of a2-dimensional slice of continuously cast steels in a billet

casting mold. The stress state in liquid region wassimulated by decreasing the elastic modulus by severalorders of magnitude as temperature increases across the

mushyzone. However, the volumetric deformation ofthe liquid region maybe developed during the 2-dimen-sional slice simulation by the mechanical constraint suchas the thermal contraction of solidifying shell and the

mold taper resulting in the development of hydrostatic

pressure which is independent of ferrostatic pressure.Kelly et al.5) developed a finite element model to

compute the thermo-mechanical state of continuouslycast round bi[lets. By striping away the liquld regionfrom strand according to the results of the heat transfercalculation, they caiculated only the stress state of so-lidified shell.

Kristiansson6) simulated a billet casting, using a finite

element analysis to determine the size of the shelllmold

gap over the entire periphery of a 2-dimensional slice ofthe strand. To describe the deformation of liquid, zerostiffness at the element above the liquidus temperaturewasassigned andzero displacement to the correspondingnodal point wasprescribed

To analyze the thermo-mechanical behavior and de-formation of liquid strand in the continuous casting

process, a finite element material model, which can easily

simulate the mechanical behavior of liquid and its effect

on the solidifying shell, is necessary.Understanding the thermo-mechanical behavior of

mushyzone during casting is very important to obtaingood quality cast products because all cracks observ-ed in continuously cast steel except transverse cracksform in this zone of low ductility.12~17) In the earlierworks.4~ Io) the simplified thermo-mechanical propertydata basedon equillbrium phasediagram wereemployedto calculate the stress at high temperature range withouttaking the deformation behavior of mushyzone into

account. However, solidification of steel during contin-

uous casting does not follow the equilibrium solidifica-

tion path due to its rapid cooling rate. Therefore, themathematical model describing the thermo-mechanicalbehavior of mushyzone as a function of temperatureand steel composition, has to be developed to calculatethe temperature and stress distributions in the solldifylngshe]1 moreaccurately, and to examine their influence onthe formation of cracks.

The purpose of this study is to propose a thermo-mechanical material model which can expiain the de-

445 O 1999 ISIJ

ISIJ International, Vol. 39 (1 999), No. 5

>-HJho:)o

Ductility lzeroducfility liquid zero

'\ ternp. impenetrable strength

Strengt~r--- temp. temp.

\ solidus\te mp.

mushy zone~

Table l. Chemical composition of carbon steel,

surrounding41moiten steej

liquidustemp.

u)H;vzaHI

Fig.

Element

Wto/o

c Si Mn P So12 o03 0.4 0,02 0,02

ZDT LIT ZST

TEMPERATURE1. Figure explaining zero ductility temperature (ZDT),

zero strength temperature (ZST) and liquid impen-etrable temperature (LIT).

Table 2. Equilibrium distribution coefficients and diffusioncoefficients of elements.22)

Element ks/L kT![ krl5 Ds (m2/s) D7(m2/s)

CSi

Mn

PS

O19

0.77

O76

O23

0.05

O34

O52

O78

O13

O035

l 79

O68

lJ)3

O57

O70

l 27x] O'~ exp(-8 1379fRT)

8.0x iO'~ exp(-248948/RT)

7.6x IO's exp(-224430!RT)

29x 10~ exp(-2301 201RT)

456xI0'4 exp(-2 146391RT)

761 xIcre exp(-1 34557,RT)

3OxIO~j exp(-25 14581RT)

5jx I0~5 exp(-2493661RT)

lJ)* I0'6 exp(- 182841!RT)

2AXlO~ exp(-223425!RT)

local equilibrium

/

Solute diffusion calculation

with FDMcomplete mixing

(a) (b)

2. (a) Schematic drawing showing the morphologyofthedendrite array and (b) the transverse cross section

assumedin the finite difference simulation.

zgHO~

10

0.8

0.6

0.4

0.2

0.0

Fig. Fig. 3.

~, i',_

/ *'1"

/ \ '~~ fs/ / sfs7fs !

/!/

l 440 1460 1480 1500 1520 1540

TEMPERATURE'C

Calculated solid fraction./;, 5-Fe fraction in solid phase~f* and y-Fe fraction in solid phase ':/:, as a function oftemperature for O. 12wto/o Ccarbon steel.

formation behavior of liquid and mushyzone. Usingthe finite element code, we have investigated the for-

mation of the air gap and the effect of the air gap onthe temperature and stress distributions in the slabcasting. The thermo-mechanical properties based on thenon-equilibrium phase diagram which is calculated withmicrosegregation analysis program have been employedinto stress and temperature field analysis.

2. Microsegregation Analysis and Yield Criterion of

MushyZone

In order to determine the solid fraction in mushyzoneas a function of temperature, the microsegregation ofsolute elements has been assessed.

Theschematic diagram and the mechanical propertiesof solidifying interface of continuous casting steel areshownin Fig. I .

Theauthors calculated the solid fractionof mushyzone as a function of temperature using thefinite difference method proposed by Ueshimall'2i,22)

which takes Into account solute diffusion in solid. Figure2(a) shows the schematic diagram of growing dendritesin the continuously cast strand. The transverse crosssection of dendrites is approximated by a regular hexa-

gon, one sixth of which is shownin Fig. 2(b). Thecom-position of steel and the thermo-physical data used in

this calculation are given in Tables I and 2, respectively.

Figure 3showsthe calculated solid fractions as a function

of temperature during solidification of the given steel.

Calculation wascarried out at a cooling rate of 0.17 K/sand at a dendrite arm spacing of I OOO~m,ll'20) In thefigure, f, is solid fraction, ~f* and vf~ represent ~-Fe andy-Fe fractions in the solid phase, respectively. Thetemperature at which the steel is fully solidified i.e.

ZDT,wascalculated to be 1450'C, which is about 43'C10wer than the solidus temperature given from Fe-Cbinary equilibrium phase diagram,

In order to describe the thermo-mechanical behaviorof the mushyzone betweenZDTand ZST, the conceptof yield criterion23~28) for porous metals was used,

The relative density, critical relative density and yield

stress of porous metal in the yield criterion have beenreplaced by solid fraction f*, critical solid fraction 'f~

and yield stress Yf, of the mushyzone, respectively.

Figure 4showsthe relative strength of mushyzone as

a function of solid fraction for various carbon steelsil'l9)

The relative strength is defined as the ratio of strengthof mushyzone to strength of fully solidified steel. Ex-perimental data show the linear relationship betweenthe solid fraction and the relative strength in the rangeof 'f~ ~f* ~ I at various carbon contents. From this re-sult, the proposed yield criterion for mushyzone gives

a satisfactory result. The critical solid fraction, 'f~, is

determined to be 0.6491 by the best fitting of the mea-sured data as shownin Fig. 4.11)

@1999 ISIJ 446

ISIJ International, Vol, 39 (1999). No. 5

>)~

IHOzLLl

oc:

H(,)LLl

>HJLuo~

Fig. 4.

lO

0.8

0.6

o.4

02

I 006c e 013c

A o.18c D 027c +o o.41C A 0,60c I+ 053c X 082c e

>{)

rl I+iinear Fltting '~ A~A+

c] AC:I

OO

cfs \ eJ~~A H:h

OO 10O.804 06O.2

SOLIDFRACTION,fRelative strength of carbon steels as a function of solid

fraction.

3. SpeciaL Procedure for Liquid Region

At the temperature above ZST, the strand in con-tinuous casting process should be taken as liquid re-

gion which should have a special property for a 2-

dimensional slice of strand. The special property in-

cludes that the stress state in liquid region should bethe pure hydrostatic stress state, which is equal to the

current ferrostatic pressure, and be independent of vol-

umetrlc deformation in the liquid region. The vol-

umetric deformation of liquid region during finite ele-

mentanalysis is inevitably developed by the thermal con-traction of solidifying shell and the mold taper In the 2-

dimensional slice method. Figure 5shows a schematicview of the volume change of the strand during con-tinuous casting in a 2-dimensional slice of strand dur-ing the finlte element ana]ysis. At the beginning of so-lidification denoted as 'A' in Fig. 5, all the regions ofstrand are liquid which has zero ferrostatic pressure. Inthe intermediate stages of solidification denoted as 'B'

and 'C' in Fig. 5, the strand Is solidified in part. Thethermal contraction of solidifying shel] and the large

mold taper give rlse to the reduction of area in the

2-dimensional slice of strand.

The stress state of strand is prescribed to a purehydrostatic pressure equal to the current ferrostatic

pressure whenthe temperature is higher than the ZST.Theferrostatic pressure, PF, acts on the interface betweenliquid and solidifying shell and can be expressed asfollows

PF=pl9z........

..........(1)

where pl' 9 and z are the density of liquid steel, theacceleration of gravity and the distance below the

meniscus, respectively. Since the distance below the

meniscus, z, changes only with the time in the finite

element analysis at constant casting speed, the ferrostatic

pressure is a function only of time at the constant pl'

At a time t+At, the basic equation to be solved in thenonlinear analysis of general structural mechanicsusingfinite element method is expressed as follows.29)

t+dt t+At ..........(2)R- F=0 .....

where the vector ' +AtR stores the externally applied nodalloads, and t+dtF Is the vector of nodal forces that are

447

Mold

2D Siice Section ,A

fB

'C

NaFrowSide Mold Taper

Wide Side Mold TaperFig. 5. Figure explaining volume reduction of liquid in

continuous casting process

equivalent to the element stress dependent only on the

current ferrostatic pressure given by Eq. (1) in liquid

reglon. Andcan be expressed as follows.

t+1ltj~1

v

BT1:dV ..........(3)F= ~_

j

Wheremis the numberof isoparametric e]ement, T is

the stress vector and the displacement-strain matrix Bis

defined by

te =BTtU.......

..........(4)

in the finite element. Here, t8 is strain vector.

Theequations used in the modified Newton-Raphsoniteration are expressed as follows

AR(i ~ i)

= t +1ltR- t +AtF(i~ 1) ..........(5)

'+1itK(i-1)AU(i)=AR(i-1).......

..........(6)

t+AtU(i) = t+AtU(i~ 1) +AU(i)..

..........(7)

with t +dtU(o) = tU ; t + AtF(o) = tF

where superscript i indicates the numberof iteration. Inaboveequations, t+AtU is the displacement vector at time

t+At and the tangent stiffness matrix t+dtK in liquid

region is expressed as follows

t+AtK =J~l fvBTCLBdV ..........(8)

where CL is the which has componentof constant value.

In this study, the componentsof CLmatrix are madeupof elastic constants of steel at melting point.5,47)

In Eqs. (5) and (6), the componentsof the vector ofnodal forces t+AtF and the stiffness matrix t+AtK areconstant over the iterations in liquid region, because the

element stress and the componentsof CLmatrix are set

to ferrostatic pressure and constant values, respectively,

in liquid region at time t+At. Therefore, the non-1inearequations to solve becomelinear equations in liquid

region. And, the strain vector te is uniform for all finite

elements in liquid region, because the componentsofstiffness matrix are constant for all elements in liquid

region.

Theaboveproposed procedure is checked against thefinite element solution of 2-dimensional hollow cylinder

problem. Figure 6(a) shows the finite element meshfor

C 1999 ISIJ

ISIJ lnternational, Vol. 39 (1999), No. 5

Elastic material elements

3 mm

7 mm

O

ll

P.^

If

Pout

(a)

OOO p.~t

OOOOOOOOO Fig. 6.

O Initial finite element meshes for calculating (a)~~~~

OOOOOOOOOOOOO~ hollow cylinder submitted to uniform pressures oninner and outer surfaces and (b) cylinder filled by

Liquid elements (P*.) the liquid elements.

(b)

~~C,,

CDUJO(

hU)

JC3

O(

la

O-1a

-2a

-30

*40

-50

-60

-70

-80

.ga

-100

-1la

fOsec

5sec

\Caleul~ted result based an Fig

Calcul~ted resul[ based on F:g

iO sec \DISTANCEFROMCENTEROFCYLINDERmm

Fig. 7. Calculated radial stresses in the cylinders based onFigs. 6(a) and 6(b) •as a function of time.

hollow cylinder of 14mmin inner diameter and 20mmin outer diameter under unlform pressures on the innerand outer surfaces. Figure 6(b) shows the finite elementmeshin which the hole of cylinder is filled by the liquid

elements to use the above proposed scheme. In this

calculation, the uniform pressure on the Inner surface ofhollow cylinder in Fig. 6(a) and the interna] pressure in

liquid elernents in Fig. 6(b) are set as follows

Pi~=10t (MPa)

where Pi~ and t the pressure and the time, respectively.

The uniform pressure on the outer surfaces of hollowcylinders in Figs. 6(a) and 6(b) is set to 50 MPa. Thecalculations are carried out under plane strain condition.

The cylinder is assumedto be elastic material whoseYoung's modulus and Poisson's ratio are 100GPaandO.3, respectively.

Figure 7showsthe distributions of radial stress alongwith the distance from center of cylinder at O, 5and 10

sec. The finite element solutions based on Figs. 6(a) and6(b) are perfectly same. It indicates that the newpro-cedure for liquid region can simulate the ferrostatic

pressure independent of deformation of liquid as shownin Fig. 5 due to the large mold taper and thermalcontraction of solidifying shell in continuous casting

process. In the analysis of continuous cating process,the solidlfied shell continuously grows with casting time.

Therefore, to apply ferrostatic pressure on solidified shell

at each time step as shownin Fig. 6(a), domain underconsideration should be constructed into finite elementmeshat each tlme step, which complicate entire analysisprocedure. With proposed method, ferrostatic pressureboundary condition can be imposed on growing solidi-

fying shell with simple procedure and no sacrifice of

accuracy.

4. Flow Stress and Thermal Expansion Coefficient in

CarbonSteel

In order to obtain an expression for the flow curvesof the carbon steel at various temperatures and strain

rates, the following constitutive equation wasproposedby Hanet a/.28)

~p=Aexp( - Q/RT)[sinh(pK)] l/"'............

(9)

with Yo= Ke'p'

where A, pand mare constants, R is the gas constant,

Qis the activation energy for deformation, K is thestrength coefficient, ep and ~p are the effective plasticstrain and the effective plastic strain rate, respectively,

and n Is the strain hardening exponent. Table 3gives theparameters for y-steel and 5-steel obtained by non-1inearfitting methodbased on the experimental data.30,31) Inthe case of ~-steel, the strain hardening is neglected.

A history of plastic deformation is stored in thematerial by a certaln arrangement of dislocations and it

seerns reasonable that this history should be lost whenthe material undergoes a phase transformation. Thus, if

a solidifying steel undergoesa ~h, transformation duringdeformation, then the accumulatedeffective plastic strain

of ~-steel becomeslost after 8/y transformation. In this

study, the effect of phase transformation on the plasticstrain was taken into account.32)

In this study, the effective thermal expansion coefficient

was calculated as follows

c( -(V/V..f)1/3 1 ..........(lO)

T- T**f

where V,.f and Vare the specific volume of the materialat the reference temperature T*.f and temperature T,

respectively. Kim et a/.21) adopted a solid fraction If,

Cc) 1999 ISIJ 448

ISIJ International, Vol. 39 (1999), No. 5

Table 3, Ptlrameters in Eq (9) for ~-ferritic and 7-austenltrcsteels.

~(MPa~1)I1 A(sec~1 ) Q(kJlmol) lvi

9997* I07~ferritic l'e O0522 2a2 1 a2657OO

Taustcnlllc I*e OO1308 O42S9 1047* IOlo .,6 . O2()08

Table 4. Spccifrc volume of ~-ferritic, '*-austemtrc Indliquid steels.35) The units of speciiic volume,temperature and carbon content are cm3/g, Kandweight percent, respectively.

Table 5. Physical data ofmold cooling water.39,

Property Value

1-lydraulie Diameter of~Caoling Channel, D O025 mFlol~* Velocity of Water

,ulv 2315m/s

Specif~lc Heat of Water, Cp** 4178 JlkgK

Tllermal Conductivity of Water, kll O614 W/mKViscosity of Water, uhv 792 x Ia-6 Ns/m2

5-fen'itic steel

Y-austenitic steel

llquid steel

Speci fic Volume

O1234+938 x10'6 (T-

293 )O, 1225 +945 x

10-6 (T- 293 )+7688 xla~e (C)

1/7035

between the solid fractions at ZDTand ZSTand thecorresponding temperature at which the solid fractionreaches T/; was defined as the liquid impenetrable tem-perature (LIT) as shown in Fig, l. It has been knownthat the so]idifying steel begins to behave like solid dueto no liquid feeding33) between dendrite arms whenthesolld fraction becomeslarger than a solid fraction ofT ' 18 34) Thus the reference temperature for the steel,/,. '

,

could be chosen to be LIT. In this study, Tf, is assumedto be 0.8.21.33) The specific voiumes for liquid, b~ and" steels were obtained by Wray's data35) as given in Table4.

5. Calculation of Heat Transfer

The temperature distribution in the transverse s]ice ofstr'and with unit thickness is c'alculated using Eq. (1 l)

for the 2-dimensional transient heat conduction accom-panying the liqLrid/solici transformatlon

i~i=p

~) aT~l a (i

"aT aT

+pLef

pC ....

c k('}-

'\' /+~'T \'t ~~~ r~'

'( 11)a)c ~ '. ., al (~!

.cT

where Tis the temperature, k is the thermal conductivity,Cpis the heat capaclty, p is the density and L is the latentheat. In this study, the enthalpy method36) was used tosolve the solidification. The initial and boundary con-ditlons are as follows

T=To,

~k,,aT

cl ••••••••••(12)c**n

where To is the Initlal casting temperature of moltensteel, ,7 is the directlon normal to strand surface and cl~

is the heat flux on the surface. The equation is solvedusing the finite element method.28) The following as-Sumptions are used in this calculation.

(1) The heat conduction in the casting direction is

negligible comparedto the heat flow to moid.(2) Theeffect of convective heat flow in liquid reglon

is taken into account using the effective thermal con-ductivity, k.rr, for molten steel.16,37) k(T) is the ther-

mal conductivity of liquld steel at temperature T.

k.ff=k(T)[1+6(1-./;)2].....

..........(13)

(3) Theheattransferbetweenmoldandcoolingwateris characterized with the ald of a heat transfer coefficient,/7w' determined from the fol]owing dimensionless cor-relation.38)

=h~D

0.023 ~ ~ (14)p~u~D)o, 8 Cp~/t~ )o 4

k~, pt~ k~

where D is the hydraulic diameter, u~ is the velocity ofcoollng water, kt,~ is the viscosity of cooling water andh~ is the heat transfer coefficient between mold andcooling water. Thermo-physical data to calculate /7~ aregiven in Table 5.39)

The thermal boundary condition between the solidify-

ing shell surface and the mold wall is modeledusing theinterfacial heat transfer coefficlent, hT, whlch is a functionof air gap thickness and surface temperature of strand.The interfacial heat transfer coefficient can be expressedas fo]10ws

/7T= l/RT+h**d........

..........(15)

where /1**d is the heat transfer coemcient for the radiativeheat flow when the air gaps occur between stand andmold and RTis the thermal resist'ance between the strandsurface 'and the mold wall except radiation. The heattransfer coefficient for radiative heat flow, h**d, maybeexpressed as foliows.

/7**d=(78(T*+ T~,)(T~ + T~'-,)

................(16)

where (T is the StefanBoltzmann constant, 8 is theaveraged emissivlty of the shell and mold surfaces, T, is

the temperature of the shell Surface and T~ is the

temperature of the mold wall. The averaged emissivityof the shell and mold surfaces, ~, was assumedto be0.44.40) The thermal resistance, RT, maybe expressedas follows

RT=Rl+R2+R3+R4.................(16)

Thecontact resistance betweenthe mold and the moldflux fiim, R1, is glven by R1= l/ hl, wherehl Is the contactheat transfer coefficient at the mold surface,41) whlchwas set to 3000W/m2K. The resistance to conductionthrough the air gap, R2, is calculated by R2=d2lk2, wherek2 is the thermal conductivity of the air and c!2 is thethickness of the air gap which is calculated from thethermo-elasto-plastic stress analysis. Theconductivity ofthe air39) was set to 0.1 W/mK. The resistance to con-duction through the mold fiux fiim, R3, is calculatedby R3=d3lk3, where k3 is the thermal conductivity ofthe mold flux and c!3 is the thickness of the gap fi]1ed

wlth mo]d flux. Theconductivity of the mold flux is used

449 '.t^, 1999 ISIJ

ISIJ International, Vol. 39 (1 999), No, 5

Cooling water channels

Wide side mold

ooa

~ oOoooONarrow side mold

strand

x

ooooooo o oCooling pi es 1--

Narrow face taper

Fig. 8.

Initial finite element meshfor calculating tempera-ture and stress of slab and temperature of mold.The characters. A, B, C and D, indicate thepositions where the variation of temperature withdistances below the meniscus Is observed.

Table 6. Temperature dependencyof heat transfe]' coeffi-

cient between mold flux and strand surf•ace 11]

h4, W/I112K

-~ Wide Side

~1'emperature Description Temperature, OK

Mold Flux Crystalline Temperature

Mold Flux Sofienlng Temperature

Metal Solidus Temperature

Metal Llquidus Temperature

l303

1423

1723

l796

1ooo

2000

6000

20000

Narrow Side

Table 7. Thermal conductivity, specific hcat ,and latent heatof carbon steel.5.43)

(a) 100 mm

Speci~c r-leat4~]

(JlkgK)

(b) 350 mm

l~emperaturc Tilermal COnduCLivityj'

(oK) (W!mK)273

473

973

l 373

l768

l796

l 823

3a

30

30

,-5

27

27

377

377

554

659

670

673

673

* I.atent r-leat orSolidification : 272 kJ/kg

to be l.OW/mK and the thlckness of the mold flux wasset to IOOptm from the mold flux consumptionanddensityof mold fiux.42) Thecontact resistance between the moldflux and the steel shell is calculated by R4=l/114, where/74 is the heat transfer coefncient between the mold flux

and strand surface. /14 mustbe dependenton temperaturedue to the large change in viscosities of mold flux overthe strand surface temperature range. The temperaturedependencyof h4 is given in Table 6.

6. Application to Continuous Casting of Slab

Figure 8 shows the initial finite element mesh for

calculating the temperature and stress of the 1600x220mmslab and the temperature of the Cu0.1010Agmold. A quarter section was modeled using symmetrycondltions. The finlte element caiculation for stress

analysis was carried out at the plane strain condition.

The thermal conductivlty of the Cu0.loloAg mold wasassumedto be 380W/mKs).Table 7gives the thermalpropertles5.43) of the strand as 'a function of temperature.Calculatlon wasperformed at the conditions of a narrow

(c) 600 mm (d) 770 mm(Mold Exit)

Fig. 9. Delbrmedgeomet]'ies of' slab and formation of air gapat various distances below the meniscus under thefriction coefnclent of 0.0 between the solidifying shell

surface and mold wall.

side mo]d t'aper of I.5 o/o/m, a casting speed of I m/min.The dlstance from meniscus to the mold exit is 0.77m.The temperature at meniscus was taken as the tem-perature which is superheated by 20'C such as 1543'C.In order to investigate the effect of contact friction

between solidifylng she]1 surface and mold wall on thedeformation behavior of strand, the calculation wascarried out at the Cou]ombfriction law with the fric-

tion coemcients of ,1=0 and 0.2.

Figures 9 shows the deformed geometries of slab at

various distance below meniscus at the frlction coeffi-

clents of /t=0. The deformed geometries of strand aremagnified by 5 times to see the formatlon of air gap.The air gap was formed on both wide and narrow side

corners in early stage of solidification as shownin thesefigures. The wide side corner of the solidifylng shell

separated awayfrom the mold wall and alr gap became

~ 1999 ISIJ 450

ISIJ International, Vol. 39 (1999), No. 5

larger during the solidification. However, the air gapnear the narrow side corner disappeared with the reduc-tion of the s]ab width by the narrow side mold taper. Theslight depression found at the off-corner of the narrowface as shownIn these figures. These trends correspondwith the severe wear of the narrow side mold which hadbeenobserved by other researchers.44'4s) Figure 10 showsthe observed geometry of the solidlfying shell, which wasobtained from the break-out shel] at a dlstance be]owmeniscus of 200mm.45)As shown in this figure, theshape of solidifying shell are in good agreernent withthe calculated results that the slight depression occurs atthe off-corner of the narrow face.

Figure ll shows the temperature distributions atvarious distance below meniscusas the slab movesdownthrough the mold at pt=0. The temperatures of 1523,

l 507 (ZST), 1500 (LIT) and 1450 (ZDT)'C correspondto the temperatures at which the solid fraction becomeO.O, O.6491, 0.8 and 1.0, respectively. In the initial stageof solidification as shown in Fig. 11(a), the uniformsolidifying shell formed on the whole surface of the slabdue to the goodcontact betweensolidifying shell surfaceand mold wal] at the given frictlon conditions. However,as solldification proceeded, the shrlnkage of the cornerregion forms the air gap that can be accompanledby

l cm

Fig. lO. Experimental shape of break-out steel shell duringslab casting at a distance below meniscus of 200mm.45]

hot spot on the off-corner regions of both the narrowand wide side surfaces as shown in Fig. 11(b). The air

gap reduced heat flow from strand to mold and gave rise

to a thinner shell in these regions which has beenobservedby previous researchers.4s -47)

It wasreported that all cracks observed in contlnuous-ly cast steel originate and propagate along the interden-drites in mushy zone except transverse crack.1217]The ductillty loss of the mushyzone Is associated withthe microsegregation of solute elements at solidifyingdendrite interfaces.14'48) This solute enriched regionremalns as liquid state resulting in decreasing of the ZDTof the steel. The dendrites which have liquid film in Its

arm space are vulnerable to tensile strain. Thus, thesolid fraction at which cracks form, should be defined toInvestigate this phenomenonin the range of 0 l.

Clyne et al.3 3) proposed the crack susceptibility coefficient

to estimate the cracking tendency in continuously caststeel. Theydivided the mushyzone into the liquid feeding

zone and the cracking zone. Cracks which are formed inthe liquid feeding zone are refilled with the surroundingllquid, whereascracks formed in the cracklng zonewouldnot be refilled with the liquid because the dendrite armsare compactedenoughto resist feedlng of the liquid. Kimet a/.ll'21) proposed that the threshold solid fraction is

Tf; and the temperature at which the solid fractionreaches T~ is defined as the LIT as shownIn Fig. l. Asliquid filrn betweendendrite armspace disappears in thetemperature below ZDT, a temperature range, TB, maybe defined as ZDT T13 The range TB indicatesthe temperature range of cracking zone defined by Clyneet al.33) In thls study, since ZDTand LIT are assumedto be the temperatures where f.= I and Tf~=0.8, re-spectively, TB is defined as 1450'C T13 1500'C. AsshownIn Fig. 11(b), the region whosetemperature is inthe range of 1450'C TB 1500'C was broader at hotspots on the off-corner regions of both the narrow andwide side surfaces than other regions at the middle stageof castlng in the mold. It indicates that the possibllity ofcracking in these regions is larger than that In other

(a) 100 mm (b) 350 mm

Unit = 'C

A=1523B=1507C=1500D=1450E=1250F=180G=160H=140I= 120J=100

Fig. Il.

Calculated temperature contours in slab and moldat various distances below the meniscus under thefriction coefficient of 0.0 between the solidlfyingshell surface and mold wall.

(c) 600 mm (d) 770 mm(Mold Exit)

451 O1999 ISIJ

ISIJ International, Vol. 39 (1999). No. 5

tlnf0,lOOB+07

o. IeQ:+e?

0.800r+Qe

O.,OOE+0G

0.888E+OO

1,1,n'

o.147!+O7

o, I IoH+07

O•794E+Oe

O•,e?E+Oe

o.SSeH+OO

(a) 350 mm, u= 0,0 (b) 770 mm(Mold Exit), kt = 0.0

1,/ml

o. IzoE+1)?

e.TO•H+oo

0.400E*OIS

o•,$4E+OO

(c) 350 mm, /~:= (Mold

,Vml

Q,lee!+07

O.104E+oe

o.$$SEtee

o.a81E+•1,

0.2 (d) 770 mm Exit), u = 0.2

Fig. 12.

Calculated heat nLIX vectors in slab andmold at v'arious distances below the

meniscus, under the rriction coemcients ofO.O and 0,2 between the solidifying shell

surfzrce and mold w'all,

OLuo(:)

Ho::

UJCL~~Lu

H

lr)I_I]

1500

1400

13ao

12CIO

1,1 rJO

1l]l]O

PosltiOn APos tion B

•-•-Pos t on C--- Posltlon D

O 20a 800600~00

DISTANCEBELOWMENISCUS,mm(a)

OO::

:)

HO(LLI

~H

1500

~500

14ao

13ac

1200

1i oa

10ao

///

lel*_O

Posltion ,e~

Pasit on 8

- - Posltl~In r_

-- Pes tlon D

l-

Wlde side shell, ,i=D O

- -Narrow side shell,~=0 O•-•-- Wlde slde 9hBll, ~=D2

\ - - Narrow slde sheil,~=D 2

[)

Fig. 14.

OUJe(~)

HO::

LIJ

O_~:UJ

H

IsOO

1400

i3c]a

12Ii. a

l I OO

10ao

,,

~'

O

Fig, 13.

200 8006aa400

DISTANCEBELOWMENISCUS,mm(b)

Variation of temperature with dlstance below the

meniscus under lhe friclion coefrilclents of (a) 0.0 and(b) O2bctwcen thc solidifying shcll surfacc and moldwall. The calculated data are observed at variousposilions in Fig. 8.

20 ~a 60 ~oo80

DISTANCEFROMCORNERmmCalculated temperature distribution of slab surface

ne'ar slab corner at a dislance below the meniscus of500mmunder the il'iction coefflclents of OOand O.'_

between the solidifying shell surface and mold wall.

regions due to retarded heat flow cause by the formationof air gap between strand and mold.

Figures 12(a) to 12(d) show the heat fiux vectors at

350mmbelow meniscus and the mold exit as the slab

movesdown through the mold at /t=0 and O.?_. Themold temperature adjacent to the corner and off-cornerreglons was lower than other parts of mold, because theair gap reduced the heat fiow from strand to mold asshownin Figs. 12(a) to 12(d), while the mold temperatureadjacent to narrow side center of strand at which the

heat transfer wasvery large, reached about 130'C at the

moid exit. Whenthe corner of solidifylng shell contactedthe narrow side mold wall due to the narrow side moldt'aper and the corner rotation,44) the heat flow from strand

to narrow side mold in this region increased as shownin

Figs. 12(b) and 12(d) and the decreasing rate of tem-per'ature in the mold adjacent to this region conse-

(.c*) 1999 ISIJ 452

ISIJ International, Vol. 39 (1999), No. 5

Unit : MP*

0.0O.51.0

1.5

~.0(il ) (b)

quent]y decreased.Figures 13(a) and 13(b) show the variatlon of tem-

perature of various positions in strand surface as shownin Fig. 8 wlth distance below meniscus at pt=0 and0.2, respectively. The surface temperatures of positions

A and D in Fig. 8 decreased as the solidification

proceeds and reached about 1100'C at the mold exit.

Thesurface temperatures of the off-corner reglons at thepositions of Band C in Fig. 8decreased more slowlythan those at the positlons A and D and the hot spotformed on these positions up to distances below meniscusof about 730mmat /t=0 and about 550mmat ,t=0.2,respectively. The slower cooling rate in positions BandCresults frcnTl the air gap formed between the mold andsolldifying shell at a distance below meniscus of about40mmunder the given friction conditions. However, thesurface temperature of position Cdecreased very fast

from dlstances below menlscusof about 730mmat kL =Oand about 550mmat ~=0.2, respectively. These phe-

nomenaresult from the contact between the narrowside corner of solldifying shell and the narrow side moldwall due to the corner rotation and the narrow side moldtaper. This contact increases the heat flow from strandto mold as shownIn Figs. 12(b) and 12(d).

Figure 14 showsthe temperature distributions of strandsurface along the distance from the corner of slab at thedistance below meniscusof 500mmat /s =Oand 0.2. Thehot spots formed on the wide and narrow side shells atabout 20mmfrom the strand corner. Figures 15(a) and15(b) show the calculated maximumprincipal stressdistributions in the solidified shell at the distance belowmeniscus of 500mmat kt=0 and 0.2, respectively. Thecalculations indicated that a tensile stress of about 2 to

3MPadeveloped at about 20 mmfrom the corner onboth surfaces of the wide and narrow faces. These10cations were very close to the hot spots as shown in

Fig. 14. Themaximumprlncipa] stress in the locatlon at,t=0.2 is higher than that at /s=0. Thls maximumprincipal stress may encourage longitudinal surfacecracks. Thus, the mold flux of high friction coefficient

can cause longitudina] surface cracks at the corner ofslab.45)

7. Conclusion

Atwo-dimenslonal thermo-elasto-plastic finite elementmodelfor the slice of strand in continuous casting processhas been developed. The model incorporates the effect

of microsegregation of solute elements on hot tears and

Fig. 15. Calculated maximumprincipal stress

contour in slab at a distancc belowmeniscus of 500mmunder the rriction

coefrilcient of (a) OOand (b) 0.2 be-

twecn thc solidifyin_g shell surfacc andmold wall.

includes a llquid modei, which can conSlder the ferrostatic

pressure due to gravity force. The thermo-mechanicalbehavior of solidifying shell of slab wasanalyzed duringthe solidlfication in the mold. The calculated reSultS oftemperature hiStOry. StreSS diStribution and formation ofair gap betweensolldlfying shell and mold in continuouScasting prOcess of slab were analyzed. The computedtemperature distribution and the principal tensile streSSdlStrlbution well explained the fonTlation of hot SpotSand cracking at Off-corner regions. The distance belowmeniscLIS Wherethe slight depresslon found at Off-COrnerregion of the narrow face becolTle shorter with Increasingthe friction coefficient between SOlldifying shell Surfaceand mOldwall. The hot spotS are formed near the wideand narrOw slde corners. TheSelOcations are very close

tO the large tensi]e Stress regions.

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