MA IERIAlS
S IEM EIl
NGIM RIMe
A
ELSEVIER
Materials Science
and
Engineering A 413414 (2005)
3743
www.elsevier.comllocate/msea
A model for prediction
of
cracks in a solidifying shell
A
Lagerstedt
*,
H
Fredriksson
Casting
o
Metals Royal Institute
o
Technology SE1oo
44
Stockholm Sweden
Received in revised form 25 August 2005
Abstract
A model coupling temperature and stress calculations with cracking criteria has been developed in order to predict crack positions in a solidifying shell. The model
is
based on a onedimensional FDM approach suitable for continuous casting
of
slabs. The strain/stress model is based on a purely elastic analysis
of
a solidifying shell giving a straightforward comparison between stresses and crack criteria. This approach makes the model easy to use. The model is numerically evaluated using available material data for
Fe2 Ni
with primary ferrite solidification and FelO Ni with primary austenitic solidification. The results
of
the calculations are discussed and the impact
of
material behavior as well as process parameters is evaluated. Evaluation
of
the influence
of
changes in the heat transfer coefficient shows that the rapid changes introduce stresses large enough to induce crack formation in the solidifying shell.
©
2005 Elsevier
B Y
All rights reserved.
Keywords:
Cracks; Stress; Solidification
1.
Introduction
Halfway cracks often appear in continuously cast material as a result
of
the combined effect
of
stress buildup during the solidification process, and low strength
of
the newly solidified material. During solidification, stresses are induced in a solidifying shell due to uneven shrinkage
of
the metal. The level
of
the induced stress depends on the cooling shrinkage and the temperature gradient in the shell. In a continuous casting process, the strand is withdrawn during the solidification subjecting the strand surface to different cooling conditions as it leaves the mold, enters different water cooling regions, etc. The reheating
of
the strand surface due to decreased cooling has been shown to influence the stress distribution in the solidified shell so that tensions arise in the inner parts
of
the strand
[13].
External loads are another source for increased stresses in a solidifying shell. External loads include the influence
of
bulging, roll misalignment and bending and straightening
of
the strand. Such phenomena have been investigated by several studies
[3 6].
Corresponding author. Present address: SSAB Oxeliisund AB, SE613 80 Oxeliisund, Sweden. Tel.: +46 ISS 254181; fax: +46 ISS 255541.
Email
address:anders.lagerstedt@ssabox.com (A. Lagerstedt).
09215093/

see front matter © 2005 Elsevier
B Y
All rights reserved. doi: I 0.1016/j.msea.2005.09.oI 7
In order to predict cracking in a solidifying shell, the thermomechanical properties
of
the material in question have to be known. High temperature tensile testing
of
in situ solidified samples gives information about the tensile strength and at what temperatures a material behaves brittle or ductile. Such measurements show that brittle behavior may be found well below the solidus temperature for some materials [7,8]. However, existing models are based on the assumption that the cracking may only occur while a liquid is present
[912].
In addition, the stress or strain in the solidifying material has to exceed a critical value
[
].
The knowledge
of
thermo mechanical properties from tensile testing together with calculations
of
temperatures and strain and stress distribution give possibilities to predict the actual location
of
the cracks in a solidifying strand, not only to identify areas where the risk
of
cracking is increased. Such a model for prediction
of
the location
of
cracks in continuously cast slabs will be presented in this study.
2
Mathematical model
2.1. Thermal model
A onedimensional analysis
of
the temperatures
in
a continuously cast strand has been performed. The solution technique applied
is
a finite difference method used with central
38
A.
Lagerstedt,
H.
Fredrikssonl Materials Science and Engineering A
413414
2005)
3743
Fig.
1.
Illustration
of
the computational grid.
approximation. A schematic illustration over the position
of
the calculation is shown in Fig.
1.
The onedimensional approach may be used as long
as
the geometry
of
the crosssection fulfills the condition that the length is much longer than the thickness
of
the strand, which may be expressed as
L»
according to the notations in Fig.
1.
The temperature calculations are based on the Fourier equation in one dimension,
a
a
(
aT
p p
= 
+q
at
ax
ax
1)
where
p
is the density,
p
the heat capacity and
K
is the heat conductivity. The source term,
q
describes the heat, which is generated within the material. Assuming equilibrium solidification, the source term is expressed, { 0,
whenT>
h
_
dgs
a
q

p( lH)
dT
at
when
Ts
<
T
<
h
0,
whenT
Ts
2)
where the equilibrium value
of
the latent heat

IN),
is distributed over the solidification range.
hand
Ts
represent the liquidus and solidus temperatures, respectively. The rate
of
solidification may be expressed in several ways. Here, the change
of
the fractionsolidified material is calculated by the expression given by Rogberg [13].
(3)
Eqs.
1) 3)
are combined in order to find the temperature distribution in the solidified layer.
2.2. Strain model
The temperature calculations are used
as
the starting point for the stress calculations, and are not coupled with the stress calculations. The derivation
of
the elastic strain is based on an approach made by Kristiansson [14]. He derived an analytical expression for the elastic strain based on the conclusion that the elastic strain was not much different from the total strain in a solidifying shell. The derivation disregards the ferro static pressure at the solidification front and assumes no remelting. The expression is valid as long
as
the coherent part
of
the mushy zones does not meet at the centreline
of
the strand. The elastic strain is expressed [14],
4)
t
is the time at which the solidification front,
XS,
passes a certain position. The thermally induced tension,
8th,
is expressed,
th
a
a
at at
5)
where
a
is the linear expansion coefficient. The strains in the solidifying shell are thereafter converted to stress by means
of
Hooke s law. Hooke s law will only give valid results within its limits
of
validity, which is below the yield stress.
2.3. Crack prediction model
A crack will form when the stress in the brittle areas
of
the material exceeds the strength
of
the material. In ductile areas, the stress will be relaxed by plastic deformation when it reaches the level
of
the material strength. Hence, in order to form cracks two conditions have to be fulfilled. Firstly, the temperature in the solidified shell has to be higher than the transition temperature from brittle to ductile state,
TDB.
Secondly, the stress in the material has to exceed the rupture limit
of
the material,
a
rI.
These cracking criteria
of
a material are found by performing hot tensile testing on in situ solidified specimens. There are several such experiments made on different steel grades, showing a sharp ductile to brittle transition temperature and
an
ultimate tensile strength gradually decreasing to zero at the coherence temperature [7,8,15]. These investigations also show that the transition temperature is affected by the cooling rate as well
as
the strain rate. The calculated stress profile is connected to the thermomechanical properties
of
the material, which is schematically shown in Fig.
2.
If
the stress in the solidifying material exceeds the measured maximum tensile strength from the tensile tests and this occurs at a temperature above the ductile to brittle transition temperature, a crack will form. This procedure is illustrated
as
path A
in
Fig.
2.
On the other hand, if the stress reaches the maximum strength
of
the material below the transition temperature the material is ductile and will therefore deform plastically, see path B in Fig.
2.
Combining the results from the calculation
of
the temperatures and stresses in the solidifying shell with the crack criteria gives the positions in the solidifying strand where the stress and temperature are sufficient for crack formation.
A.
Lagerstedt,
H
Fredriksson
/
Materials Science
and
Engineering
A 413414
2005)
37 43
39
Ul Ul
en
Stress
?
'\
I
I \
, .........................................
7
...
l
//
I
~~ ~
..
Temperature
t
x
B
I
S L
T
DB
Temperature
Fig.
2.
Schematic illustration
of
how cracking criteria is combined with stress profile to determine crack position.
3.
Calculations
3.1. Data used in calculations
The theoretical model presented above were used in order to investigate how material dependent data, such
as
coherence temperature and brittletoductile transition temperature,
as
well as process dependent parameters such as the heat transfer coefficient influence the cracking behavior
of
a continuously cast slab. The model is evaluated for the two binary alloys Fe2 Ni and FelO Ni. The material data used is summarized in Table
1
n
cases where actual material data is not available, values for delta ferrite has been used for Fe2 Ni, and values for austenite has been used for the FelO Ni. Studies
of
how the change
of
a specific parameter influences the results are made starting from the base data presented in Table
1
Unless otherwise stated by reference indication, the material properties have been estimated with aid
of
the commercial software IDS Solidification Analysis Package.
Table 1 Material properties used in the calculations Property Symbol Liquidus temperature
DC)
h
Solidus temperature
DC)
Ts
Coherence temperature
DC)
Tcoh
Brittleductile transition temperature
DC)
TD
Heat conductivity
of
solid (W/(m K))
ks
Heat conductivity
of
liquid (W/(m
K»
kL
Heat capacity
of
solid (W/kg K)
c
P
s
Heat capacity
of
liquid (W kg K)
C
L
P
Density
of
solid (kg/m3)
Ps
Density
of
liquid (kg/m3)
P
Equilibrium heat
of
fusion (kJ/kg)
t.H
Elongation coefficient
(I/K)
a
Elastic modulus (GPa)
E
Thermomechanical properties
of
Fe with 2 and
10
Ni have been evaluated in high temperature tensile testing on in situ solidified samples by Hansson and Fredriksson [8]. These values are adopted in this investigation and are included in Table
1.
It is
of
interest to note that the transition temperature for
Fel
O Ni
is found approximately 200 K below the solidus temperature while the ductile to brittle transition in Fe2 Ni
is
found to be less than 50 K below the solidus temperature. In the same investigation the coherence temperature was found to be 151O°C for Fe2 Ni and between 1465 and 1490 °C for
Fel
O Ni
[8], which is somewhat below the solidus temperature for both alloys. In the present study, the temperature under which the material can withstand strain will be set to the solidus temperature. However, in order to see its influence on the stresses, a temperature corresponding to a fraction solidified
of
0.6, according to Eq. (3), and the measured coherence temperature will also be used. During a solidification process, the heat transfer coefficient normally varies
as
a function
of
time. This is pronounced in a continuous casting process as the strand moves from one cooling zone to another. In order to investigate the impact
of
a change
of
heat transfer on the cracking, calculations have been performed for cases with a constant heat transfer coefficient
of
800 W/(m
2
K)
throughout the solidification process as well as cases where the heat transfer coefficient is changed. In cases with a varied heat transfer coefficient, the change is set to occur after 120 s and different decreases
of
the heat transfer coefficient has primarily been tested. The aim is to illustrate the changes occurring during a continuous casting operation
as
the heat removal drastically decreases
as
the strand leaves the mold. The shell thickness has been set to 0.1
m,
the casting temperatures was
10K
above the liquidus temperature and ambient temperature 35°C.
3.2. Results
of
calculations
The cooling curves for the two alloys Fe2 Ni and FelO Ni are shown in Figs. 3 and
4,
respectively. The figures show the results in three positions; the surface, the centreline and a point midway between surface and centre. In the first 120
s,
Fe2 Ni
1528 1524 1524 1500 35.2 35 738 813 7287 7057 242 22 x
10
6
70
FelO Ni
1503 1497 1497 1300 36.7 35 698 798 7368 7139 260 24
X
10
6
70 Remark Ref.
[i7]
Ref.
[i7]
Assumed Ref. [8] Est. from Ref. [18]
40
A
Lagerstedt,
H
Fredriksson
/
Materials Science
and
Engineering A
413414
2005)
37 43
1600r.. r
1400
E
1200
~
~
~
1000
Q
0
800
I
600
Solidification interval
HTC
= 200
¥'
_HTC=600
HTC
=
700
.::::::.
HTC
=
800
HTC
=
900
400~=====c======c=~
__ ______ __
o
200 400 600
800 1000
Time[s]
Fig. 3. Cooling curves for
Fe2%Ni
at surface, centre and a point between the two. Calculations with different changes
of
heat transfer coefficients are shown.
1600r..~~.__
1400
~
~
~
1000
~
E
~
800 600
~~~ HTC
= 600
~HTC
= 700
HTC=800
~HTC=900
.__. Solidification interval
400~==============~ ~ ~
o
200
400
600
800
1000
Time[s]
Fig.
4.
Cooling curves for FeJO Ni at surface, centre and a point midway between the two. Calculations with different changes
of
heat transfer coefficients are shown.
all calculations have a heat transfer coefficient of 800 W/(m
2
K
and it
is
thereafter changed resulting in temperatures
as
shown. Figs. 5 and 6 show the temperature profiles after 150 and 500 s for the two alloys. The stress resulting from three different coherence temperatures and a constant heat transfer coefficient are shown in Figs. 7 and 8 for the two alloys.
In
the figures, stress profiles
1500
E
1400
Q
a
1300
Q
0
E
1200
Q
I
No HTC change HTC change to 700 HTC change
to
600 HTC change to 400 HTC change to 200 Solidification interval
0.1
Fig. 5. Temperature profiles for
Fe2%Ni
30
and
380
s after change
of
heat transfer coefficient.
1500 1400
IT
~
1300
a
1200
0
E
~
1100
HTC
change to 900
No
HTC
change
HTC
change to 700 HTC change to 600
HTC
change to 400
HTC
change to 200
Solidification
interval
0.1
Fig. 6. Temperature profiles for FeJO Ni
30
and
380
s after change
of
heat transfer coefficient.
50r~   ,
~
p
~~
I.()
en
0
cn
LO
U
0
/)
~g
u ;g
~o
1 9
g
1 0
II
;
II
;
II
II
II II
50
::...JI..::=='I=' ':::''I~====:__......:....I...:7':~hl.....:::........_=_
o
0.02 0.04 0.06 0.08
0.1
Distance from cooling surface [m]
Fig. 7. Calculated stress distribution in
Fe2%Ni
with different coherence temperatures.
for calculations with constant heat transfer coefficient for the times
150
and 500 s are shown. Figs. 9 and
10
show the stress distribution resulting from a change
of
heat transfer coefficient for the Fe2 Ni. The calculated stress profiles are shown for the times 150 and 500
s.
The ultimate tensile strength for the temperature distribution
50 .. .
~
~
lOcn
Den
Orn
OU
~9
~o
0
~o
....
LO
~
g g
II
~~
II II II II
50~~_=~1~=:  1~~J~1~~~1~~
o
0.02 0.04 0.06 0.08
0.1
Distance from cooling surface [m]
Fig.
8.
Calculated stress distribution in
Fe
\o%Ni
with different coherence temperatures.
A.
Lagerstedt,
H
Fredriksson
/
Materials Science and Engineering A
413414
2005)
37 43
41
100
Rupture limit HTC change
to
900
TC change
to
200 0.01 0.02 0.03 0.04
Distance from cooling surface
m]
Fig. 9. Calculated stress distribution in
Fe2 Ni
after 150 s with different heat transfer coefficients.
100
HTC change to
900
o change HTC change to
700
HTC change to
600
HTC change
to
400
Rupture limit
0.01 0.02 0.03 0.04
Distance from cooling surface
m]
Fig.
10.
Calculated stress distribution in
Fe2 Ni
after 500
s.
corresponding to the case with a constant heat transfer coefficient
of800W m
2
K) is also shown. Stress profiles for the FelO%Ni alloy are shown in the same manner in Figs.
11
and
12.
During each time step, the positions where the calculated stress exceeds the ultimate tensile strength and the temperature at the same time exceeds the ductile to brittle transition temperature
ro
150
100
a
50
~
o
HTC
change
to
600
Rupture limit
0.02 0.04 0.06 0.08
0.1
Distance from cooling surface [m]
Fig.
11.
Calculated stress distribution in FeIO%Ni after
150s
with different heat transfer coefficients.
HTC change
to
200 ';;;' 100
HTC
change
to
400
'
HTC
change
to
600
~
0
1
::om~~t~~~:~~~~:~ ~~R:u~p~ture
limit
<Jl
~
n
~
100
co
'S
C,
r
0 200
300~ ~ ~ ~ ~~ ~
o
o ~
QW
QOO QOO
0.1
Distance from cooling surface
m]
Fig.
12.
Calculated stress distribution in FeIO%Ni after 500
s.
o~~..  .
100 200
~
Liquidus front
OJ
300
E
i=
400
500
600~
__ __
7
__
____
__
~
o
0.02
0.04
0.06 0.08
Distance from cooling surface
m]
Fig. 13. Predicted cracking (shadowed area) in
Fe2 Ni
with constant heat transfer coefficient
of
800 W/(m
2
K).
Coherence and ductile to brittle transition temperatures are 1524 and 1500°C.
are recorded. The result for the entire solidification process is plotted for some performed calculations in Figs. 1316.
The
grey surface in these figures shows the time and position in the shell where the conditions for cracking are fulfilled.
o
200
~' . '''' ''
~~.
~
400
OJ
E
i=
600 800
o
0.02
~
.~.(iqUidUS
fronl
.
.
~
..


Solidus front
\.
~\.
0.04 0.06 0,08
0.1
Distance from cooling surfce [m]
Fig, 14. Predicted cracking (shadowed area) in
Fe2 Ni
with heat transfer change from 800 to 200 W/(m
2
K
after 120
s,
Coherence and ductile to brittle transition temperatures are 1524 and 1500
DC.