Simulation of solidification and viscoplastic stresses during vertical semicontinuous direct chill casting of aluminum alloy

International Journal of Plasticity, Vol. 8, pp. 161-183, 1992

0749-6419/92 $5.00 + .00 Copyright © 1992PergamonPress Ltd.

Printed in the U.S.A.

SIMULATION OF SOLIDIFICATION AND VISCOPLASTIC STRESSES DURING VERTICAL SEMICONTINUOUS DIRECT CHILL CASTING OF ALUMINUM ALLOY

T. INOUE and D.Y. Ju KyotoUniversity (Communicated by

Otto Bruhns, Ruhr-Universit/itBochum)

A b s t r a c t - A method to simulate temperature and stress fields in solidifying cylindrical ingots

during the vertical semicontinuous direct chill casting process is presented in this paper. As the first approximation of the whole process, a steady state in the casting ingot with enough length is examined, and the steady heat conduction equation with heat generation due to solidification incorporated with material flow is solved by the finite element method. Based on the obtained temperature distribution, elastic-viscoplastic stresses were also calculated to obtain residual stresses. Here, material parameters appearing in the constitutive equation were determined by the data of tensile stress-strain curves under some strain rates at several temperature levels close to the melting temperature. Calculated temperature and residual stress fields were compared with the experimental data to verify the analytical procedure, and discussions are also given on the effect of operating conditions on the fields.

I. I N T R O D U C T I O N

Vertical semicontinuous direct chill casting process recently developing is one of continuous casting methods to produce ingots of aluminum alloys and other metals without long mold. It is beneficial for optimizing the operating condition to succeed in the simulation of thermomechanical field in the solidifying ingot. So many reports have been published concerning analysis of the temperature distribution incorporating solidification by finite element method, but a few papers treat the induced stress-strain field: 2-D simulations of thermal stress in continuous casting slab were made by using elastic-plastic constitutive model (GRILL et al. [1976]; SORIMACHI& BRIMACOMBE[1977]), and viscoplastic stresses were simulated based on the solidification analysis (WILLIAMSet al. [1979]). The aim of this paper is to apply the method of analysis of temperature and stresses incorporating solidification developed by the authors (WASG& INOUE [1985a, 1985b]) in the normal continuous casting process (INoUE [1987]; hqOUE & WANG[1988]; WANG& INOUE [1986]) to semicontinuous direct chill casting of aluminum alloy. In the initial stage of the casting process, when the bottom block plate is located at the upper position (see Fig. 1) and the length of the growing ingot is small, the temperature, liquidsolid interface, and stresses in the ingot vary with time, both in the sense of space and of material. However, when the ingot becomes long enough, the fields in the upper part are regarded to be time-independent or steady in the spatial coordinate fixed to the system. In the first part of this paper, a steady heat conduction equation with heat genera161

162

T. INOUEand D.Y. Ju

Molten metal ~ ~

.:.......:.~: : ...

Refractory

'

Cooling water Mold W a t e r flow

f

J J

~

/ / /

l!l

~ /

Billet

Bottolll block

/

Withdrawal V Fig. 1. Schematicview of semicontinuousvertical direct chill casting system.

tion due to solidification is formulated in a spatial coordinate system when considering the material flow. A numerical calculation for the temperature in the solidifying ingot as well as the simulation of the location of liquid-solid interface is carried out by finite element technique. Most metallic materials at low temperature may be treated as an elastic-plastic solid. However, if they are heated beyond the melting point, the materials can be regarded as viscous fluid, and they behave in a time-dependent inelastic manner at high temperatures dosed to the melting temperature. Therefore, a unified constitutive model is needed to be established to describe the elasto-plastic and viscoplastic behavior of the solidified part of the ingot as well as the viscous property of the liquid state. Taking into account the effects of such phenomena, a modification of Perzyna's constitutive model is presented in the second part of the paper, and some experimental results of the viscosity appearing in the model are presented for AI-Zn type alloy employed. By using the model, elastic-viscoplastic stresses are calculated in the ingot to know the residual stress distribution, and are verified by the measured data from the hole-drilling straingage technique. Finally, results of numerical simulation are presented on influence of operating condition on temperature and stresses , such as ingot size, casting speed, initial temperature and so on, to give a fundamental data for optimizing the operating condition.

Simulationof solidificationand viscoplasticstresses

163

H. GOVERNING EQUATIONS

II. 1. Coordinate system and material-time derivative In the continuous casting process, the material flows in space, so that the fields, for example, temperature and stress, can be generally described in a spatially fixed Eulerian coordinate system x instead of a Lagrangean coordinate X being co-moved with the material with velocity V. In this case of casting, the material velocity V is assumed to be a sum of the casting speed v and the small displacement rate ~ different from v. Then, the material-time derivative of any time-dependent scalar, vector, or tensor function ~b(x, t) is reduced to

a~ ~ = -~- + V-grad 4,

(1)

or, in an axisymmetric cylindrical coordinate system (r-z), which will be used in the following analysis;

a¢,

a~

a¢,

~,= - ~ + V r ~ r + V z ~ zz.

(2)

In the following, the equations are presented in the general coordinate system, but the reduced forms in the cylindrical coordinate system are abbreviated for simplicity.

II.2. Heat conduction equation incorporating solidification Suppose that tho heat generated by mechanical work (INo~E [1987]; I~Ot~E ~ WAnG [1985]) is quite insignificant compared with the heat capacity involving in the ingot. An uncoupled equation for heat conduction including latent heat generation Is for volume fraction of solidifying element ~s is expressed in the form (Isoo~. ~, WAnG [1988]) pc~r - div(k grad T) - p l ~ = 0,

(3)

where p, c, and k are, respectively, the density, specific heat, and heat conductivity of the material. Henceforth, all material parameters X are assumed to be expressed by mixture law (Bow~N [1976]) such that X =Xs~sq-Xl(1 --~s),

(4)

with the parameters Xs and X, for solid and liquid. When the terms of 0( )/Ot in eqn (1) vanish for steady state, eqn (3) is now reduced to pcV-grad T - div(k grad T) - p/sV-grad ~s = 0.

(5)

164

T. INOUE and D.Y. J u

In this problem, the boundary conditions of temperature can be written as T = ~

on St,

(6)

and -kn.grad T= 0

on Sq,

(7)

or

-kn.grad T= h(T-

Tw)

on Sh,

(8)

in which ~ is a fixed temperature on the boundary St, and h and 0 are the heat transfer coefficient and the heat flux on the boundaries Sh and S a with unit normal n, while Tw is the temperature o f coolant. To identify the volume fraction of solid phase ~s during solidification appearing in eqns (4) and (5), use was made of a phase diagram. Figure 2 shows the diagram of an AI-Zn alloy employed in this study, in which T~ and rn~ are the liquidus temperature and the gradient on liquidus line if#, and T~ and ms are related to solidus line ~ . When neglecting the in-solid phase transformation from o~-phase to (o~ + #) at the point f , the fraction of solidifying phase is determined by the formula of lever rule as

( T ~ - T)/ml ~s = [ ( T _ 7~,)/ms + (T~ - T)/m~] '

(9)

6O0 ~ 400

d .L+ p •

;

\p

c~ 200 ~-

0 0 A[

20

40

60

80

Atomic percent of Zn % Fig. 2. Phase diagram of A I - Z n type alloy.

100 Zn

Simulation of solidificationand viscoplasticstresses

165

and the volume fraction o f liquid ~Jl is given as (10)

~l = 1 - ~s. 11.3. Equations for stress analysis

We can assume the deformation of the body to be characterized by the infinitesimal strain theory. The rate type equation of motion for the Cauchy stress a is written as div 0 + p6 = O,

(11)

in which b is the body force, and the Eulerian infinitesimal strain rate i is given by the spatial gradient of the deformation rate fi as

~)T}/2.

i = [grad fi + (grad

(12)

The boundary conditions for the rates of displacement and stress in steady state read fi=60

onSu,

(13)

and # . n = ~0

on

St,

(14)

where n is the unit vector normal to the traction boundary St, and 60 and ~0 are the prescribed displacement and traction, respectively. The total strain rate i is given by the sum o f the elastic and inelastic strain rates ie and i~, as well as the thermal strain rate i r and the strain rate im representing dilatation caused by phase change. i = ie.at- i i - t - iT.~- i m

05)

where, the elastic strain ee is

~e _

1 -t- ~,

e

a -

~,

(tr a ) l .

(16)

Here, E and ~ denote the Young's modulus and the Poisson's ratio depending on temperature. The constitutive relation of the thermal strain rate ~ r is

i t = a~'l,

(17)

and the transformation strain rate due to solidification yields i m = fl~sl,

(18)

166

T. l~otrE and D.Y. Ju

where a and/3 denote the coefficients of thermal expansion and of dilatation due to solidification, respectively. In order to formulate the inelastic constitutive equation relevant to describing the behavior of solidifying material from liquid to solid, a unified viscoplastic model of Perzyna type (PERzYNA ~ WoJr~O [1968]) is employed ii_~ " 1

OF

3# 0-~'

(19)

with the viscosity # and the modified static yield functions F (WANG & INOUE [1985a])

(20)

F = f ( o , T) - oz.

Here, the linear hardened static flow stress oy = oy(T, ~-i) = Oy0(T) + H'(T)~ -i

(21)

is employed, where, H ' is the hardening coefficient with the equivalent viscoplastic strain ~i and Oyo stands for the initial yield stress depending on only temperature in solid state and being constant in liquid, and F


=J0

; F;

F< 0

(22)

F>O.

If we employ the Mises type function f ; f=

(3J2) 1/2,

J2 = ½trs2,

(23)

eqn (19) is now reduced to

1(

ii= ~

1

(3J2) 1/2

s,

(24)

where s stands for the deviatoric stress. In the case of uniaxial stress state, eqn (24) is duplicated as 1

ii= ~

( o - Cry).

(25)

This means that the viscosity #(T) as well as static flow stress oy(T, gi) can be identified if conventional tension test data under different strain rates at several temperature levels are provided as shown in Fig. 3. The calculated value of the viscosity is represented in Fig. 4 depending on temperature. Since the temperature-dependent value of/~ in so-

Simulation of solidification and viscoplastic stresses

167

T = I00 "C 600

~ = IO0%/min 450

b

- 300 10

u1 ,,~

I

~50

0

0.1

1

|

0

4 Stroin E , °1o (~)

~ = I °/o I m i n

600

450

0

- 300 O

t

200

m vl ~_

N ~50

2~0

l 0

i

O.

,i

2

4

~train

E

.

°1o

Co) Fig. 3. Stress-strain diagrams of AI-Zn type alloy: (a) strain rate dependence; (b) temperature dependence.

168

T. |NOUE and D.Y. Jv

m

10000

~. :~

:2

8000

~-~ •" : ' -

~n o ~.9 q~

6000

"9 4000' ~-

0

0 .~

c-

0

2000

o~

u

.~

~0

00

~)

I

I

I

1 0 200 300 400 Temperoture T,~C

500

Fig. 4. Coefficient of viscosity determined.

lidifying liquid state is hard to be identified, the constant value was assumed by extrapolating the measured ones in solid state beyond 346°C.

!11. P R O C E D U R E OF NUMERICAL CALCULATION

III. 1. Finite element equation for temperature analysis The finite element formula of heat conduction equation is obtained by applying the Galerkin method to eqn (5) with the boundary conditions (6)-(8); ([P] + [H] + [Sh])[T I

----- [Q~]

+ [QhJ

+

[Qq].

(26)

Here, temperature T in an element is characterized by the temperature vector IT ]e on the nodes by an interpolation function IN] as T = [N] IT] e.

(27)

The first term of the right hand side of eqn (26) represents the latent heat generation due to solidification, and the second and third terms correspond to the heat flows through boundaries Sh and Sq, respectively. The terms in eqn (26) are obtained by summing up the following element matrices for the entire domain, which are generally:

[P] = e~

fvOC[NlrlVlr[Gl

dV,

(28)

Simulation of solidification and viscoplastic stresses

169

EH~=e~ L *~°ITE°I~V'

(29,

E~J= e~ f~ hENITENI~,

<30,

h

[Q:} = ~ ~M~{~rad ~}r[V}

[N]rdV,

[Q~}=~ £ hT.[N]rdS,

(31)

(32)

h

~.' : - ~ 2 o ~ s .

(~,

q

Here, [G] is the matrix of gradient operator of [GI = [ V I [ N I III.2.

r,

[N], shown as

with[Vl()=grad().

(34)

Finiteelementequationfor stressanalysis

When the vector of displacement rate ~~ I and strain rate ~i I in a finite element are interpolated by the nodal displacement vector ~ Ie in the forms lal = [NI lal ~, and

~il = [B] lal ~,

05)

with the strain-displacement matrix [B] dependent on coordinates, the system of stiffness equation in incremental form is derived from eqns (11)-(15) by applying the principle of visual work as [K]I~I = ~Lbl + lLrl + ~L~I + IL~I.

(36)

The terms in the above equation are composed by summing u~ the element matrices generally shown as follows:

fv [B]r[De][B]dV, [L~] = ~ fv [~]r([b}[O])[V]dV' [K]=

~

[Lr] : ~

fv [B]r[De]{S}
(37)

(38)

<39)

170

T. l~otn/and D:Y. Ju

[ L . , ] = e~

fv[B]r[De][~l(~[~])lV}dV'

[L~l=-~ fv(O~)[B]r[D~]lsldV,

(40)

(41)

with [~] = [ ~ ] [ ) ~ ] .

(42)

Here, T ~ and ~ are the temperature and the volume fraction of solid phase at the central point of the element. [K] denotes a total stiffness matrix, and [D e] represents the elastic stress-strain matrix, while [ s I and [ V ] are the vectors of deviatoric stress and casting speed. Other vectors [&], {~} and scalar ¢o are

[<~l = [c~l - 0 ~ [ D ¢ ] - ~ [ ° l '

(43)

0 1~1 = [~31 - ~ [ D e ] - ~ [ o l , 0~

(44)

~ : ~

111.3.

l(

1

(3jz)l/z "

<45)

Calculation scheme of numerical analysis

In the processes incorporating phase transformation, the coupling effect between the fields of temperature, material phase, and stress (and strain) are essentially to be taken into account in the analysis (Ir~ouE [1987]). As stated in the previous section, however, the numerical analysis of steady temperature distribution coupled with solidifying phase change was made independently followed by stress analysis based on obtained temperature field. Here, an assumption is made on the velocity field in both analyses that the casting velocity v is large enough compared with the displacement rate ti in this case of uniform casting, which follows that the velocity [ V } in eqns (28)-(40) is to be replaced by the uniform casting speed {v I. Solving the finite element eqn (26) by neglecting the first term [Qsl on the right hand side representing latent heat generation in the first approximation, liquids and solidus line are determined. By use of the lever rule, the volume fraction of solid ~s is identified, and the term I Qs } is evaluated. The same iteration procedure was repeated to obtain the converged distributions of temperature and phase. This obtained result is applied to determine the displacement rate vector i~l in eqn (36), and the displacement u, strain ¢, and stress ~ can be evaluated by integrating the rate/~, i, and b. Moreover, the eqns (26) and (36) derived in the previous section are nonlinear with respect to the temperature I T ] and displacement rate I ~ ], respectively, and special attention is necessary in solving the equations. Since the equation involves only one term

Simulation of solidification and viscoplastic stresses

171

for latent heat generation determined by volume fraction o f solid ~s, it is enough to employ the simple method o f iteration. As for stress analysis, on the other hand, the nodal forces on the right hand side o f eqn (36) are the nonlinear functions o f temperature T and structural change ~s being dependent on the rate [~1. Furthermore, in the region o f molten state, a large strain rate is essentially produced in spite o f small variation o f nodal forces. A modified Newton-Raphson scheme, or initial stress method, was adopted to solve the nonlinear equation. IV. RESULTS OF NUMERICAL CALCULATION AND EXPERIMENT

The theory and the procedure developed above are now applied to the simulation of the vertical semicontinuous direct chill casting process schematically shown in Fig. 1. The material treated is A1-Zn type alloy with 5.6e/0 zinc and 2.50~0 magnesium. A quadrilateral finite element mesh pattern o f 600 elements with 1941 nodes illustrated in Fig. 5 is employed for both analyses of temperature and stress fields. The boundary condition for heat conduction is assumed in such a way that the temperature of the meniscus o f molten metal is prescribed to be To, and that heat is insulated along the central line and the bottom of the ingot as well as the surface contacted with the refractory. The cylinder facing the mold is regarded as the boundary Sq on which heat flux is given, and the other part of the surface Sn is cooled by water of temperature Tw. Figure 6 depicts the measured heat flux t~ absorbed by the mold, and heat transfer coefficient h depending on flow rate o f water W~ is shown in Fig. 7. Other data used for temperature calculation incorporated with solidification are: Heat conductivity:

k~ = 0.0425 ks = 0.0125 + 0.0583T

cal/(mm.°C)

Density:

p = 2.8

g/ram 3

Specific heat:

Cl = 0.015 + 0.00173T Cs = 0.083 + 0.00267T

cal/(g.°C)

Latent heat due to solidification: is = 93.16

cal/g

Casting speed:

v = 80.0

mm/min

Liquidus temperature:

T~ = 638

°C

Solidus temperature:

Ts = 600

°C

Gradient o f liquidus line: ml = 3.69

°C/070

Gradient o f solidus line: ms = 9.09

°C/%

As for the finite element calculation of stresses based on the obtained temperature distribution, the following material data are employed: Young's modulus: Poisson's ratio: Viscosity:

El = 500.0 Es = 75000.0 - 76.3T ~ = 0.33

MPa

# = 3700.0 MPa. s /~ = 0.007178T 2 - 21.1698T÷ 10160.7

(T_> 346°C) ( T < 346°C)

172

T. IyouE and D.Y..lu

Refractory mr

t~

t 1

\Mold

I I I

L

I

l l

I

I

1 I 1

,,

Z Fig. 5. Finite element mesh pattern.

Initial yield stress:

a~o = 2.0

MPa

( T _ 346°C) ( T < 346°C)

MPa

( T < 150°C) (150 < T < 346°C) (T>_ 346°C)

Oyo = 150.0 - 0 . 4 2 8 T H a r d e n i n g coefficient: H ' = 350.0 - 0.13333T H ' = 330.0 - 1 . 6 8 3 ( T - 150) H ' = 0.132 T h e r m a l e x p a n s i o n coefficient: og = 33 × 10 - 6 as=21.8× 10-6+0.2T× D i l a t a t i o n d u e to solidification: ~ = -7.5

10 - 7

I/°C %

Simulation of solidification and viscoplastic stresses

173

0:8

E E

0.6¸

u

0.4 0.2 -r"

I

O0

I

40

80

Distance from top of mold

120 d , mm

Fig. 6. Variation of heat flux from top of the mold.

Characteristic results of calculated temperature and residual stresses for an ingot of 1 m in length with a diameter of 240 mm are compared with experimental data to verify the method, and simulations in other cases of different operating conditions such as casting velocity, size of ingot, and cooling rate are also made.

'~E --,

O

1.6

fl-.-

U

a

.d

A

115

o

85

,~.

~~

~.2

Wl= 145 I/m[n

)

o~. o~ ~,~ ,~

I~ 0

u

~..

0.8

~,~

m 0L ~-

~ 112

0.~

•

0

100

200

300 1000

Distance from the bottom of mold d,mm

Fig. 7. Distribution of heat transfer coefficient depending on discharge of cooling water.

174

T. l~ot~ and D.Y. Jt~

~

TI

I

V = B0m r n / m i n

-: I ~ ~ ~oo~i~L / ~ ~ o o t ~ ~ l l ~ ~ ~

~

~~ o o 1 ~ ~ ' ~ ~ ~ % ~

~:~

~-

~ ~1 ~ ~~ 0

~i,~

-~°°

::~,~,"--

~.~,o~'~xs~c~

~ Fig. 8. Bird's eye view of calculated temperature profile.

IV. 1. Temperature distribution and mode of solidification Figure 8 indicates a bird's eye view and an isothermal representation of calculated temperature distribution. The lines denoted by TI and T~ in the figures are the liquidus and

800

600

V= BOmm/rain

~

2

•

P ~00

_2

I~

fi

~- 200

Surface~

0 0

o_ ~,~2~12o:2:o~

I

!

100

~

~

200

Distance from meniscus

~

~

300 z, mm

Fi~, 9. Temperature variations a[ the center and surface of the into[.

~

400

Simulation of solidification and viscoplastie stresses

~...~-:..-=p~----

1.0 O O" tO

/'

,/"

g.8

/

.,,,"/./ ; /

/

0.6

~.//~.. ~,°

~.,,,..~.~.,~_.~..:;.~

80

/"

r= 38 m m

~I .: ~ ; i

] /'

E O.2 0

,' ~

,,.;

i i

0A

r=0

/~

[

O

175

i

/

/°

.........

r=76 r=120

!

v= 80 ram/rain

"" / /

/

~

,

I

100 120 140 Distonce from rneniscu~ z, m m

160

Fig. lO. Volume fraction of solid along the distance from meniscus.

solidus temperature, respectively. If the data are replotted to give the temperature change at the center and on the surface, we have the result shown in Fig. 9, where the circles represent the measured temperature by thermocouples. The fact that the calculated temperature on the surface coincides well with the measured data may indicate the validity of the simulation method. Change of the volume fraction ~s at four characteristic points along the distance from the meniscus is represented in Fig. 10, which may give information on the progressing mode of solidification and the thickness of the solidified shell.

=

V = 80ram/rain, ~

~ 0~1

~-~0

0.0

~

~

-

?

-

~; ~~XIF ~ "~ ~ ~

~

~

~

~

~ ~

~

~

. . . ~ ~ ~,~',~

~,,,~

Fig. 11. Bird's eye view of deformation.

176

T. l~oLr~ and D.Y. Jt~

~

V

= 80 mml,n~n

~

~ sot

~ o.o

ooo Eoo

'

~

~

~

10(3

~ 0.0 ~ -100 t3

~ -u ,~

~% o.o

V

,~

= 80mmlmln

"~

g

~," ~00 ~

~

~

'

~

~o

-~00

°

~-~

~~i~/~

~ G ~

0D

[r°m ~e~isc~

9kst~ce

~ Fig. 12. Bird's eye view of stress distributions.

IV.2. Stress distribution For the simulated results of temperature and mode of solidification presented in Sec. IV. 1, analysis o f stresses was carried out for the whole area of the ingot, including the molten metal, by use of the finite element eqn (36) based on the elastic-viscoplastic constitution model (15). The displacement mode is depicted in Fig. 11. The bird's eye views and the contour o f radial, tangential, and axial stress distributions ~r, ao, az are shown in Fig. 12. The contours of equi-stresses lines are represented in Fig. 13.

Simulation of solidificationand viscoplasticstresses

V:80

0.0

177

ram/rain

Or

C~z

.\

200

/

E E N

400

O

0

E 0

/

600 •--. 0

(:~ -~"

~

~. ~

~'

800

1000

0

24

48

72

96

Radius r, mrn

120 0

24

48

72 96

Rodius r, mm

1200.

24 48

72 96 120

Rodius r,mm

Fig. 13. Iso-stress c o n t o u r s .

Examples o f the calculated stress distribution by the elastic-viscoplastic constitutive model and by the time-independent elastic-plastic model are shown in Fig. 14. When compared with each other, the viscoplastic stress analysis gives smaller results, at least on the surface, than elastic-plastic stresses. The stresses are found to generate at the location where the solidification starts (see Figs. 12 and 13), and the radial distribution going downstream becomes steady owing to the flatter temperature distribution. In order to examine this effect, the stress distributions at several locations are given in Fig. 15, in which the distributions at the very end of the ingot (Fig. 15d) can be regarded as residual stresses. Circles in the figure indicate the experimentally measured residual stresses by a hole-drilling strain gauge method (REDNER & PERRY [1982]). The fact that the experimental data coincide well with the analytical results suggests the validity of the simulation procedure based on the viscoplasticity developed here. As seen from Fig. 15, the shear stresses are insignificant in the area of molten state, and the normal stresses ar, tr0, and a z (see Figs. 12 and 14) are regarded to be hydrostatic stresses, which means that the constitutive equation employed here reveals to modelize the liquid.

178

T. INOUEand D.Y. Ju

V= 61)mm/rnin

0.0

de

~

2OO

~%

-~ ~

~00 i

~

60o

t~

~S 800

10000

24 z~

72 9~

120 0

N

z~ 72 96 120

~cK~iu~ r, mm (a)

(b)

Fig. 14. Calculated circumferential stresses by elastic-viscoplastic model (a), and elastic-plastic model (b).

2OO ((3) z=121mm

(b)

z=4OOmm

IO0 0.0

~ ......

" ~ ~

~~

~'~.~.-.~-'7_~__~ -~=

O EL

~ -100

~ -200 _

=lO00mm

6

~

1oo

~

0.0

i.~

~2%-.-,~---

-10(3 -200

0

24

48 72 96 .1200 24 Radius r, mm

48

72 96 120

Fig. 15. Stress distribution in several sections of ingot.

Simulation of solidification and viscoplastic stresses

179

. V=80 o

N

o m U U~

c

250 ~-

1o~

1~

500

0

120 0

120 0

120

Rc~dius r, m m (a) 0.0

I1

V=70

o

v=60

0

o

E ~.00

HO

E

o

U

C 0

HO

OC

0

150 0

250

Radius

0

300

r, mm (b)

Fig. 16. Isothermal lines in ingot: (a) effect of casting speed; (b) effect of ingot radius or cooling water rate.

180

T . l ~ o u E and D.Y. Ju

E 38 E cO N >~

0 ff~ m I~ r~¢-

. _

.12 I--

v=80 mm/min

36 34 32 3O 8O

I

T

100

I

120

i

140 150

Discharge of cooling water w~, tlmin

Fig. 17. Relation between thickness of mussy zone at the center depending on discharge of cooling water.

IV.3. Simulations for other operating conditions Let us apply the procedure to other operating conditions. Hereafter, focus is placed on the effect of temperature and stress distributions on the casting speed, size of ingot, and cooling condition. Figure 16 (a and b) represents examples of temperature profiles for different casting speed and the radii of ingots with various cooling rates. The effect of discharge of cooling water on thickness of the mussy zone at the center is summarized in Fig. 17. The effects of casting speed and ingot radius on the stress distributions are represented in Fig. 18 (a and b), respectively, and the relation between stresses and casting speed is summarized in Fig. 19. The simulated stresses varying with the discharge of cooling water are also plotted in Fig. 20. The calculated results shown above seem to simulate the characteristic temperature and stresses depending on operating condition. If we accumulate the data of such simulation at different operating conditions as shown in Figs. 16-20, the possibility of optimizing the design of the system would be high.

V. C O N C L U D I N G REMARKS

A method to simulate solidification and temperature as well as stress distribution in vertical semi-continuous direct chill casting process is formulated, and the implementation by finite element calculation is presented in this paper. A modification of Perzyna's viscoplastic constitutive relationship was proposed, which reflects actual liquid-solid phase transformation during the casting process. The results of simulation are also verified by comparison with experimental data, and application of the technique to other operating conditions is carried out to obtain the fundamental data of optimum design of the system.

Simulation of solidification and viscoplastic stresses

181

0.0

~E 200 d .~ t, O0

E

~ 600 o=

;5 800

O.

2~, z-8 72

96

120 0

Radius r, mm

24 ~8 72

96

120 O

Radius r , m m

2& ~8

72 96

t20

Radius r,mm

(a)

O. Wt=112 I/min

;;

E E

Clz

_ .~_ ~=14 W 2l .

r Wt= 245 -

~.

~

~ ~--.

~-,~

r-

500

;.

E

E

o

. ,(.~ ~.

c}

~_ t

1000

.I

,,~/ /

I

.

O.

150 0 250 0 Radius r, mm V=80 mm/min (b)

300

Fig. 18. Stress distributions: (a) effect of casting speed; (b) effect of ingot diameter.

182

T. INouEand D.Y. gu

E ._- 50 m

r=120mm wt=115mm

45

o__

0 m

t-

/ //

~0

o~

~I

m m I~

a5 30

~,. ~.~

~

25 55

70 85 Casting.speedv, mm/min

100

Fig. 19. Relation between stresses at the center of ingot and casting speed.

~ 60 V=80

mmlmin

Z= 112 rnm

07 ~ 40

"C~ .I/~

1 ~

.~

/

~ 20 I~ ff~

~

.~--- - ' ~

~

t

Oo

m ~_

~'

0

70

90

110

D i s c h a r g e of coating w a t e r

130

1SO

W l , I/rain

Fig. 20. Relation between stresses at the center of ingot and discharge of cooling water.

REFERENCES

1968 1976 1976 1977

PERZYNA,P., and WOJNO,W., "Thermodynamics of a Rate Sensitive Plastic Material," Arch. Mech. Stos., 20, 499-511. BOWEN,R.M., "Theory of Mixture," in ERINGEN, A.C. (ed.), Continuum Physics, Vol. 3, Academic Press, New York, pp. 1-127. GRILL,A., BRn~tACO~aE, J.K., and WnINB~RG, F., "Mathematical Analysis of Stresses in Continuous Casting of Steel," Iron and Steelmaking, 1, 38-47. SORa~ACrn,K., and BRn~ACOUBE,J.K., "Improvements in Mathematical Modeling of Stresses in Continuous Casting of Steel," Iron and Steelmaking, 4, 240-245.

Simulation of solidification and viscoplastic stresses

1979

183

WnJ~Aus,J.R., L~w~s, R.W., and MonoA~, K., "An Elastic-viscoplasticThermal Stress Model with Applications to the Continuous Casting of Metals," Int. J. Num. Meth. Engng., 14, 1-9. 1982 l~t)~R, S., and P~.RRY,C.C., "Factors Affecting the Accuracy of Residual Stress Measurements using the Blind Hole Drilling Method," in Proc. 7th Int. Conf. Experimental Stress Analysis, New York, August, 604-616. 1985 INou~,T., and WANG,Z.G., "Coupling between Stress, Temperature and Metallic Structures during Processes Involving Phase Transformations," Mater. Sci. Technol., 1, 845-851. 1985a WA~t~,Z.G., and INotr~, T., "Viscoplastic Constitutive Relation Incorporating Phase Transformation-Application to the Welding," Mater. Sci. Technol., 1, 899-905. 1985b WA~, Z.G., and I~o~-~,T., "A Viscoplastic Constitutive Relationship and the Application to Carburized Quenching and WeldingProcess," in Proc. Int. Conf. Nonlinear Mechanics, October, Shanghai, pp. 657-662. 1986 WA~, Z.G., and I~otr~, T., "Analysis of Temperature and Elastic-viscoplasticStresses during Continuous Casting Process," in Proc. Int. Conf. Computational Mechanics, May, Tokyo, pp. 103-108. 1987 I~o~J~,T., "Metallo-thermo-mechanical Coupling-Analysisof Quenching, Welding and Continuous Casting Processes," Berg- und Hiitteum~innischeMonatsbefte, 132, 63-75. 1988 I~otn~,T., and WAbt~,Z.G., "Thermal and MechanicalFields in Continuous Casting Slab-A Steady State Analysis," Ing. Arch., 58, 265-276. Department of Mechanical Engineering Kyoto University Kyoto, Japan

(Received 15 September 1989; in final revised form 8 April 1991)