On the shrinkage of metals and its effect in solidification processing

by

Anders Lagerstedt

Casting of Metals Royal Institute of Technology SE-100 44 Stockholm, Sweden

Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan i Stockholm framlägges till offentlig granskning för avläggande av teknologie

doktorsexamen fredagen den 10 december 2004 kl. 10.15 i Kollegiesalen, Kungliga Tekniska Högskolan, Valhallavägen 79, 100 44 Stockholm

ISSN-1104-7127 TRITA-MG-2004:02

On the shrinkage of metals and its effect in solidification processing

Anders Lagerstedt

Casting of Metals

Royal Institute of Technology, KTH SE-100 44 Stockholm, Sweden

Abstract The shrinkage during solidification of aluminium and iron based alloys has been studied experimentally and theoretically. The determined shrinkage behaviour has been used in theoretical evaluation of shrinkage related phenomena during solidification. Air gap formation was experimentally studied in cylindrical moulds. Aluminium based alloys were cast in a cast iron mould while iron based alloys were cast in a water-cooled copper mould. Displacements and temperatures were measured throughout the solidification process. The modelling work shows that the effect of vacancy incorporation during the solidification has to be taken into account in order to accurately describe the shrinkage. Crack formation was studied during continuous casting of steel. A model for prediction of crack locations has been developed and extended to consider non-equilibrium solidification. The model demonstrates that the shrinkage due to vacancy condensation is an important parameter to regard when predicting crack formation. The centreline segregation was studied, where the contributions from thermal and solidification shrinkage were analysed theoretically and compared with experimental findings. In order to compare macrosegregation in continuous casting and ingot casting, ingots cast with the same steel grade was analysed. However, the macrosegregation due to A-segregation is driven by the density difference due to segregation. This is also analysed experimentally as well as theoretically.

This thesis is based on the following supplements: Supplement 1 J. Kron, A. Lagerstedt and H. Fredriksson ‘Measurements and modelling of air gap formation in aluminium base alloys’ ISRN KTH-MG-INR-04:02SE TRITA-MG 2004:02 Accepted for publication in International Journal of Cast Metals Research Supplement 2 A. Lagerstedt, J. Kron, F. Yosef and H. Fredriksson ‘Measurements and modelling of air gap formation in iron base alloys’ ISRN KTH-MG-INR-04:08SE TRITA-MG 2004:08 Supplement 3 J. Kron, A. Lagerstedt and H. Fredriksson ‘Modelling of air gap formation in solidification processing’ International conference on solidification science and processing, Bangalore, India, November 17-20, 2004 Supplement 4 A. Lagerstedt and H. Fredriksson ‘A model for prediction of cracks in a solidifying shell’ ISRN KTH-MG-INR-04:06SE TRITA-MG 2004:06 Supplement 5 A. Lagerstedt, S. Adolfi and H. Fredriksson ‘Crack formation during continuous casting of tool steel’ ISRN KTH-MG-INR-04:07SE TRITA-MG 2004:07 Supplement 6 A. Lagerstedt and H. Fredriksson ‘Macrosegregation in continuously cast tool steel’ ISRN KTH-MG-INR-04:05SE TRITA-MG 2004:05 Supplement 7 A. Lagerstedt, J. Sarnet, S. Adolfi and H. Fredriksson ‘Macrosegregation in ingot cast tool steel’ ISRN KTH-MG-INR-04:09SE TRITA-MG 2004:09

Contents 1. Introduction.............................................................................................. 1 1.1 Background to shrinkage in solidification processes............................ 1

1.2 Aim and content of the thesis................................................................ 2 2. Modelling of solidification................................................................... 3 3. Air gap formation ................................................................................... 7 3.1 Introduction........................................................................................... 7 3.2 Experimental work................................................................................ 8 3.3 Mathematical model............................................................................ 11

3.4 Discussion ........................................................................................... 16 4. Crack formation .................................................................................... 17 4.1 Introduction......................................................................................... 17 4.2 Experimental work.............................................................................. 18

4.3 Mathematical model............................................................................ 19 4.4 Mathematical evaluation of the experimental work ........................... 22 4.5 Discussion ........................................................................................... 25

5. Macrosegregation ................................................................................. 27 5.1 Introduction......................................................................................... 27 5.2 Continuous casting.............................................................................. 28

5.2.1 Experimental work .................................................................... 28 5.2.2 Mathematical evaluation............................................................ 30

5.3 Ingot casting........................................................................................ 33 5.3.1 Experimental work .................................................................... 33 5.3.2 Mathematical evaluation............................................................ 35 5.4 Discussion ........................................................................................... 37 6. Concluding remarks ............................................................................. 39 7. Future work............................................................................................ 41 Acknowledgements ......................................................................................... 43 Bibliography ..................................................................................................... 45

1

Chapter 1 Introduction 1.1 Background to shrinkage in solidification processes During solidification of a metal, the density of the material changes due to cooling of the metal in both liquid and solid state as well as due to the solid to liquid phase transformation itself. Phase transformations in the solid state during solidification may also cause a volume change which will affect the solidification process. The superheated liquid metal cools to the liquidus temperature where solidification starts. During the cooling, the melt experiences thermal contraction due to the lowered temperature. The solidification is also connected to a density change. Depending on the solidification interval of the solidifying metal, the shrinkage may occur at a specific temperature or distributed over an interval. During subsequent cooling of the solidified material, the solid metal experiences thermal contraction. Uneven cooling of the solid induces stresses and strains in the material. Phase transformations in the solid state may contribute to the shrinkage of the solid metal. The peritectic reaction in steel is one such important phase transformation, where the transformation from ferrite to austenite is accompanied with a shrinkage. Knowledge of the shrinkage behaviour is an important parameter when modelling such parts of the solidification process as air gap formation, crack formation and strain induced macrosegregation. The formation of an air gap occurs due to the contraction of the solid shell which is withdrawn from the mould. The formation of an air gap drastically reduces the heat transport across the mould/metal interface, and thereby affects the solidification process by the decreased heat removal. Cracking stems from the low strength of the solidified material in combination with a stress build-up in the solidified shell. The stress distribution in the solid shell during casting depends to a large extent on the thermal shrinkage of the solid material. Macrosegregation occurs due to the movement of enriched liquid, which in some cases are induced by shrinkage, but are also caused by density differences arising from the microsegregation. One type of macrosegregation which is caused by the shrinkage is the centreline segregation seen in continuous casting.

2

The effects of non-equilibrium solidification have been in focus in several studies.1-5 The reason for the investigated effects of the non-equilibrium solidification has been attributed the incorporation of excess levels of vacancies into the solidified material. The increased amount of vacancies will decrease in the solid as the excess level diffuse towards, and condense in, the grain boundaries. Amongst other effects, the incorporation of vacancies is thought to affect the evolution of latent heat. The combination of discoveries of the importance of vacancy incorporation into the solid during freezing,1-5 and the previous knowledge that vacancy condensation affects the density of a metal,6,7 implies that the condensation of vacancies also may effect shrinkage related phenomena during solidification. Research regarding the shrinkage behaviour with specific aim on air gap formation in aluminium and copper has also shown that the condensation of point defects which are incorporated into the solid material during solidification is an important factor to consider when investigating the strain in a solidifying body.8-10

1.2 Aim and content of this thesis This work aims toward an increased understanding of the shrinkage behaviour during solidification and at applying the results in order to better explain some shrinkage related phenomena occurring in solidification processes. The formation of air gap during solidification has been investigated in supplement 1, 2 and 3. Air gap measurements are performed and the shrinkage behaviour during solidification is modelled and verified by the experimental data. Crack formation is a phenomenon highly dependent on shrinkage induced stresses. In supplement 4 a model is developed for the prediction of cracks. The model is based on an elastic formulation and compares the calculated stresses with the thermomechanical properties of the material. In supplement 5 the model is extended to include the vacancy condensation, proved to be an important factor of the shrinkage in supplement 1 and 2. The extended model is compared with experimental values of the crack positions. The shrinkage occurring during the final solidification of a continuously cast slab is responsible for an increased concentration of carbon and sulphur in the centre of the slab, also known as centreline segregation. The centreline segregation is discussed in supplement 6, where an analysis of the contribution of cooling and solidification shrinkage to the centreline segregation is made. Macrosegregation also occur in ingot casting, which is treated in supplement 7. The main reason of the macrosegregation in ingots is density differences in the liquid state. A contributing factor to the density difference is the increase of alloying elements in the remaining liquid during solidification and not a shrinkage of the material itself.

3

Chapter 2 Modelling of solidification Modelling of a solidification process involves consideration of the heat transports in and away from the solidifying and cooling material. Heat is transferred by conduction in the material, which is described by the Fourier heat equation:

qTktTCP +∇=∂∂ 2ρ (1)

ρ is the density, PC is the heat capacity and k is the heat conductivity of the studied material. T is the temperature and t is the time. The source term, q , describes the heat released from within the metal. During equilibrium solidification the source term is expressed:

( )

<

<<∂∂

∆−

>

=

S

LSS

tab

L

TTwhen

TTTwhentT

dTdg

H

TTwhen

q

0

0

ρ (2)

The tabulated latent heat, ( )tabH∆− , is the heat released during equilibrium solidification, Sg is the fraction solid, LT is the liquidus temperature and ST is the solidus temperature. Depending on the material studied the fraction of solid is expressed differently. When regarding steel, the change of the fraction solidified material may be calculated by the sine function given by Rogberg:11

( )

−−

−−

−=

π

π

21

2sin1

SL

LS

L

S

TT

TTTT

dTdg (3)

In aluminium the fraction solid may be described by Scheil’s equation, assuming that the diffusion of alloying elements in the solid phase is slow:

4

−−

−−

−=fk

mL

mS TT

TTg1

1

1 (4)

mT is the liquidus temperature of the pure substance, and fk is the partition coefficient. During solidification under non-equilibrium, vacancies are incorporated into the solid phase, and thereby increase the energy level of the solid. The vacancies diffuse towards the grain boundaries and condense. The condensation is accompanied by a heat release as the solid decreases its energy level toward the equilibrium state. When modelling non-equilibrium solidification, the source term in the Fourier heat equation has to include the release of heat due to vacancy condensation:

( ) ( )

( )

<∂∂

∆−

<<∂∂

∆−+∂∂

∆−

>

=

Sv

v

LSv

vS

sol

L

TTwhenMt

xH

TTTwhenMt

xHtT

dTdgH

TTwhen

q

1

10

ρ

ρρ (5)

The heat released during the non-equilibrium solidification will also decrease compared to the equilibrium solidification since the energy level does not reach the equilibrium state. The equilibrium value of the latent heat is divided in two parts were one part accounts for the heat of solidification, ( )solH∆− , and one term accounts for the condensation of vacancies formed during solidification:2

( ) ( ) ( )( )veqv

incvsoltab HxxHH ∆−−+∆−=∆− (6)

incvx is the concentration of vacancies incorporated into the solid during

solidification and eqvx is the equilibrium concentration of vacancies. ( )vH∆− is the

heat released due to condensation of vacancies. The separate term for the condensation of vacancies makes it possible to account for condensation occurring below the solidus temperature. The equilibrium concentration of vacancies in the solid phase is varying with the temperature. The temperature dependence is written:

∆−

∆

⋅=TBkvH

BkvS

eqv eex (7)

vS∆ denotes the entropy of the vacancy and Bk is the Boltzmann constant. Due to the supersaturation of vacancies in the solid phase after the solidification the fraction of vacancies will decrease toward the equilibrium value by diffusion of vacancies to grain boundaries where they condense. This diffusion process occurring in the solid state may be described by a relation used to describe

5

diffusion during homogenisation processes. The average vacancy concentration is here expressed as a squared sine wave function:

( )π

λπ

22

24

−

−+=tD

eqv

incv

eqvv exxxx (8)

D is the diffusion constant and λ is the diffusion distance. In the mushy zone the diffusion distance changes over time, as new material solidifies on top of already solidified material. In the model used in this work, the calculations of the concentration are simplified to consist of the average value between newly solidified material with the maximum vacancy concentration given by the supersaturation, and the value given by the equation 8, calculated with a characteristic length related to the fraction solidified material at that temperature. The numerical solution of the presented equations may be performed in several different ways. In this work, the heat diffusion equation was solved with a finite difference method. The model assumes constant material properties within a small and limited volume, and the thermal influence of the neighbouring volumes, as well as any internal heat release, is calculated during each time step. The time step has to be small enough so that events occurring during solidification are treated properly. Solidification over narrow solidification interval can present difficulties because of the problems to resolve the very short times where heat is released.

6

7

Chapter 3 Air gap formation 3.1 Introduction Air gap has a big influence on the heat transfer between casting and mould. There are many studies on the connection between variations in the heat transfer coefficient and the formation of an air gap.12-18 The formation of an air gap starts when the solid metal shell becomes strong enough to withstand the pressure from the melt and thereby departs from the mould due to the contraction.19 Before the formation of an air gap, heat transfer occurs mainly through conduction. When an air gap forms, the heat transfer is decreased and can be described by superposition of the radiation and conduction terms.20 For aluminium the heat transfer due to radiation is insignificant compared to conduction and could be neglected,21 while for steel it has to be taken into account. Due to the small size of the gap, convection in the gap can be neglected. The general explanation to the formation of the air gap is the thermal contraction of the solid shell. The thermal contractions arise due to the decreased temperature at the surface and resulting thermal gradients. There are, however, experimental work showing that the thermal shrinkage does not alone explain the size of the measured airgap.8,9 The mathematical treatment of the shrinkage and its contribution to the air gap formation may include modelling of viscoplastic-elastic stresses and strains,22 although some investigations claim that an elastic model is sufficient to describe the strain during solidification.23 The use of more advanced models has so far not been able to give a more accurate prediction of the size of the air gap.24 In recent work a model which includes the shrinkage of vacancy condensation has been developed and used in the examination of air gap formation in pure aluminium, eutectic aluminium-silicon and Al-Cu, and copper based alloys.8-10 The model connects the material shrinkage with the formation and condensation of vacancies as well as the thermal contraction when the alloys are cast in a cylindrical mould. The aim of the present work is to extend this model in order to obtain a more thorough description of the solidification process and the shrinkage

8

causing air gap formation when aluminium and iron based alloys are solidifying over a solidification interval.

3.2 Experimental work Casting experiments were conducted in top poured cylindrical moulds. The Al-based alloys were cast in a mould made from low carbon steel with an outer diameter of 250 mm and a thickness of 50 mm. The Fe-based alloys were cast in a water-cooled copper mould with an inner diameter of 250 mm and a thickness of 10 mm. The top and the bottom of the moulds were insulated in order to achieve a heat flow in radial direction mainly. During the casting experiments the temperature distribution in the castings were recorded. In the aluminium based experiments the temperature distribution in the mould was also measured, while in the iron base experiments the mould temperature along with the temperatures of the cooling water were measured. Simultaneously the movements of the inner mould wall and the solidifying metal were measured giving the air gap as a function of time. The heights of both moulds were 100 mm and all measurements were made at half the height. The experimental set-up used for air gap measurements of Fe-base alloys is schematically drawn in figure 1.

Figure 1: Experimental set-up for iron base experiments.

9

An example of the measured temperature distribution in ingot and mould is shown in figure 2 for an aluminium based alloy. The air gap forming in the same experiment is shown in figure 3. When the metal is poured, the mould is heated with a consequent expansion. When the cast metal has developed a solid shell thick enough to withstand the pressure of the liquid metal, the solid shell contracts and departs from the mould. At the same time, the continued heating of the mould causes a continued expansion. The air gap is the distance between the mould and the metal, and will therefore consist of both movements. The same type of curves for the temperature distribution and air gap formation in an iron base alloy are shown in figures 4 and 5. The metal solidifies quickly on the surface of the water-cooled copper mould and an air gap is seen to form immediately. The movement of the mould is somewhat different in this case. The mould is rapidly heated during the filling of the mould and an expansion can be seen in the measurement of the mould displacement. The lowered heat transfer due to the air gap formation, together with the strong cooling by the water, causes a decrease in the mould temperature and the mould consequently contracts, thus reducing the growth rate of the formed air gap. Since the castings have been insulated in bottom and top and the core gets rapidly heated to the temperature of the melt, the heat losses are very small in all other directions than the radial direction.

0 50 100 150 200 250 300 350 400100

200

300

400

500

600

700

800

Tem

pera

ture

[°C

]

Time [s]

Figure 2. Measured temperature distribution during solidification of Al-4.5%Cu.

10

0 50 100 150 200 250 300 350 400-2

-1

0

1

2

3

4

5 x 10-4

Time [s]

Dis

plac

emen

t [m

]

Casting

Mould

Figure 3. Measured displacements of casting and mould during solidification of Al-4.5%Cu.

0 100 200 300 400 500 600 700 800 900 1000800

900

1000

1100

1200

1300

1400

1500

1600

Time [s]

Tem

pera

ture

[°C

]

Figure 4. Measured temperature distribution during solidification of Fe-10%Cu.

11

0 100 200 300 400 500 600 700 800 900 1000-5

0

5

10

15

20 x 10-4

Time [s]

Dis

plac

emen

t [m

]

Casting

Mould

Figure 5. Measured displacements of casting and mould during solidification of Fe-10%Cu.

3.3 Mathematical model The solidification process was described by the numerical solution of the Fourier heat equation discussed in chapter 2. The heat equation is solved for cylindrical coordinates in the radial direction with a Dirichlet boundary condition. The surface temperature was extrapolated from the measured temperature distribution. In the aluminium base experiments the extrapolation was made by fitting a polynomial to the measured data. In the experiments with iron base alloys, the strong cooling caused an air gap to form immediately and therefore a polynomial fit could not describe the surface temperature since the surface temperature starts to decrease before the thermocouples closest to the mould experience any temperature decrease. The extrapolation of the surface temperature was therefore made by parallel movement of the time-temperature curve of the thermocouple closest to the mould. Figure 6 shows the extrapolated interface temperature together with the calculated and measured temperature at the thermocouple in the melt closest to the core for an aluminium base alloy, and figure 7 contain the corresponding measured, extrapolated and calculated temperatures for an iron base alloy.

12

0 50 100 150 200 250 300 350 400450

500

550

600

650

700

750

800

Time [s]

Cal

cula

ted

and

mea

sure

d te

mpe

ratu

re [°

C]

Calculated temperaturer = 25 mm

Measured temperaturer = 25 mm

Interface temperature

Figure 6. Measured and calculated cooling curves from experiment with Al-4.5%Cu.

0 100 200 300 400 500 600 700 800 900 1000700

800

900

1000

1100

1200

1300

1400

1500

1600

Time [s]

Cal

cula

ted

and

mea

sure

d te

mpe

ratu

re [°

C]

Interface temperature

Calculated temperaturer = 109 mm

Measured temperaturer = 40 mm

Measured temperaturer = 110 mm

Calculated temperaturer = 40 mm

Figure 7. Measured and calculated cooling curves from experiment with Fe-10%Cu.

13

In the thermomechanical models the shrinkage is described by the superposition of elastic strain and shrinkage due to vacancy condensation. Relations to calculate the total shrinkage are given in equations 9 and 10 coming from Hooke’s law and a volume balance describing the shrinkage achieved by the ordering of the atoms. The total displacement of the outer surface of the casting is then the sum of the thermal displacement, thu , and the vacancy induced displacement, vu .

( ) ( ) ( )

−++−

+−⋅= ∫ ∫r

a

R

a

TrdraR

arTrdrr

uth 22

22 111 νννα (9)

22

22aR

uRux vvv −

−⋅⋅=∆ (10)

In equation 9, α is the thermal elongation coefficient and ν is Poisson’s number. The geometry is described by the outer radius, R , the location of the solidification front, a, and the spatial parameter, r. The calculations of the thermal strain were based on the results from the modelling of the solidification process. In order to model the process of vacancy condensation and its effect on the displacement of the casting, the fraction of vacancies incorporated in the solid at the solidification front must be known. In the case of the experiments with aluminium alloys, the incorporated fractions were estimated by comparing the measured undercooling with data given by Fredriksson and Emi.2 During the evaluation of the iron based alloys, the fraction of incorporated vacancies were assumed to be 10 times the equilibrium concentration at the solidus temperature. Examples of the resulting calculated displacements of the solidifying ingots are shown for Al-4.5%Cu in figure 8, and for a Fe-10%Cu in figure 9.

14

0 50 100 150 200 250 300 350 4000

1

2

3

4 x 10-4

Time [s]

Dis

plac

emen

t [m

]

Vacancy displacement

Total displacement

Measured displacement

Thermal displacement

Figure 8. Calculated and measured displacements of the Al-4.5%Cu casting.

0 100 200 300 400 500 600 700 800 900 10000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2 x 10-3

Time [s]

Dis

plac

emen

t [m

]

Measured displacement

Thermal displacement

Vacancy displacement

Total displacement

Figure 9. Measured and calculated displacement of Fe-10%Cu casting.

15

In commercially available softwares for simulation of solidification processes, a boundary condition which usually has to be given is the heat transfer coefficient. The heat transfer coefficient may be evaluated from the temperature measurements in the experiments. The amount of heat transported over the metal/mould interface is described by equation 11, where both conduction and radiation is included.

( ) ( ) ( )( ) ( )im

imeteff

im

imet

im

imet TTAhTTATTAh

dtdQ

−=−+−=44εσ (11)

A is the contact area at the interface, h is the conduction heat transfer coefficient, ε is the emissivity and σ is the Stefan-Boltzmann constant. The superscript i denotes interface and the subscrips met and m indicate metal and mould respectively. The same amount of heat has to be transported away by the mould. In the aluminium experiments, this transport is given by the temperature gradient in the mould, as expressed by equation 12. In the experiments concerning iron base alloys the heat transported away by the mould is expressed in terms of the temperature difference and flow of the cooling water and the temperature of the mould.

i

m

drdTk

dtdQ

−= (12)

wwPw

wmmPmm TC

dtdV

dtdTCV

dtdQ

∆+= ρρ (13)

V , k , ρ and PC denote the volume, the heat conductivity, the density and the heat capacity respectively and the subscript w indicates water. wT∆ is the temperature change of the cooling water. The contribution to the heat transfer from radiation can be considered small compared to the heat transfer by conduction for the aluminium alloys,21 but not for the iron base alloys. The heat transfer coefficients are presented in figure 10. In the aluminium case, the heat transfer coefficient first shows a rapid increase, and after reaching a maximum value a decrease is seen. The first increase is deduced to come from the extrapolation of the temperate gradient in the mould, and can be ignored. In the real case the heat transfer is likely to have the maximum value from the beginning of the solidification. The decrease of the heat transfer coefficient value coincides with the measured formation of an air gap. In the case of iron solidification in a water cooled copper mould, the initial solidification is very rapid and the maximum value can not be distinguished. This is a result of the extrapolation of the metal interface temperature as well as the fact that solidification starts during the filling procedure.

16

0 50 100 150 200 250 300 350 4000

200

400

600

800

1000

1200

1400

1600

1800

2000

Time [s]

Hea

t tra

nsfe

r coe

ffici

ent [

W/m

2 K]

Al - 4.5%Cu

Fe - 10%Cu

Figure 10. Evaluated heat transfer coefficients.

3.4 Discussion Air gap measurements have been performed on several aluminium and iron based alloys. The modelling work was based on findings that vacancy condensation occur during a solidification process,1-5 and that the vacancies also influence the density.6,7 Air gap measurements performed earlier on pure aluminium, eutectic aluminium and copper have shown that vacancy condensation influences the air gap formation.8-10 Although good agreement with the experimental data was obtained, the results from the modelling work show that the influence of the vacancy condensation is less pronounced in the solidification of iron base alloys. This is attributed to the fact that the temperature gradient in the solidifying steel is larger. The reason for this is the stronger cooling since a water-cooled mould was used. Furthermore the heat conductivity of steel is much lower than for aluminium. Altogether, the air gap measurements performed on aluminium and iron base alloys together with the performed calculations including the effect of vacancy condensation clearly indicate that a non-equilibrium model better explains the formation of air gap.

17

Chapter 4 Crack formation 4.1 Introduction 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. The formation of cracks occurs in the cast material as a result of the combined effect of stress build-up during the solidification process, and low strength of the newly solidified material. The formation of half way cracks during continuous casting is one such phenomenon. In the continuous casting process, the strand is withdrawn during the solidification, subjecting the strand surface to different cooling conditions as it leaves the mould, 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.25-27 External loads, including bulging, roll misalignment, bending and straightening of the strand are other sources for the increase of stresses in a solidifying shell. Such phenomena have been investigated in several studies.27-30 In order for a crack to form, the stress or strain in the solidifying material has to exceed a critical value.25 Many existing crack formation models are based on the assumption that the brittle cracking only may occur while a liquid film is present.31-34 Experiments concerning high temperature tensile testing of ‘in-situ’ solidified steel samples have shown that the transition from brittle to ductile deformation often is found well below the solidus temperature, and that the transition temperature is dependent on the cooling rate.35-37 The studies of air gap formation during solidification discussed in the previous chapter showed that the incorporation of vacancies and the consequent condensation is an important factor to consider when investigating strains in a solidifying material. This fact makes it increasingly interesting to investigate how the consideration of non-equilibrium solidification will affect the cracking behaviour of a solidifying shell. In supplement 4, an elastic model to predict the

18

crack formation was therefore developed. The model is extended to include the effects of non-equilibrium solidification and compared to experimentally obtained values in supplement 5.

4.2 Experimental work A continuous casting experiment was carried out in slab caster 2 at SSAB Oxelösund AB. The outline of the caster is seen in figure 11. The caster is described in more detail in supplement 5 and 6.

Figure 11. Outline of the casting machine used in plant trials.

During the experiment the casting speed was changed so that two different casting speeds could be investigated in one heat. However, in order to obtain steady-state conditions, each of the two casting speeds had to be kept constant during a length equivalent to the metallurgical length. The experiment was started with a casting speed of 0.66 m/min and after 30 m of cast strand the withdrawal rate was increased to 0.78 m/min. In all, 59 m of strand were cast. A total of nine samples were cut from the strand, evenly spread along the length of the strand. The samples were etched and chemically analysed. The structures in all samples were purely columnar, and contained half way cracks. Samples cast with different casting speeds exhibited different initiation points and end points for the cracks. The locations of the initiation and ending points of the cracks are plotted as functions of the strand length in figure 12. A change in crack appearance is clearly visible in the macrostructures. This change of crack appearance is called

19

inner crack initiation, while the outer starting position is called outer crack initiation.

0 10 20 30 40 5020

40

60

80

100

120

140

Strand length [m]

Dis

tanc

e fro

m s

trand

sur

face

[mm

]

Upper sideLower side

Crack end

Inner crack initiation

Outer crack initiation

Figure 12. Crack locations as function of cast length.

Thermomechanical properties for the tool steel cast in this investigation were determined by high temperature tensile testing of in-situ solidified samples. The solidification interval of the alloy was determined by thermal analysis. The continuous casting experiment was, apart from the crack investigation presented in supplement 5, also used as base for an analysis of the centreline segregation presented in supplement 6.

4.3 Mathematical model The base in determining if, and where, crack formation occurs is the comparison between the calculated stress profile and the strength of the material. The stress in the solidifying material has to exceed the strength of the material given by the temperature at the position. At the same time, the material has to be in a temperature interval where it is brittle. Hence, in order to form cracks two conditions have to be fulfilled. Figure 13 illustrates the connection between the measured thermomechanical properties and the calculated stress profile. The left side of figure 13 represents the measured properties of the material, while the right side of the figure consists of

20

the calculated stress distribution. If the stress in the solidifying material exceeds the measured ultimate tensile strength 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 figure 13. 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 figure 13. The temperature distribution in the solidifying shell was calculated with the Fourier heat equation in one dimension with Cartesian coordinates as discussed in chapter 2. The computational grid is illustrated in figure 14.

Figure 13. Schematic illustration of the connection between stress profile and cracking behaviour.

21

Figure 14. Schematic illustration of the calculation grid.

Kristiansson studied the stress and strain in a solidifying material.23 He derived an analytical expression for the elastic strain in a solidifying shell and concluded by comparing with numerical models of the strain, that the elastic strain was not much different from the total strain in a solidifying shell. The expression derived by Kristiansson23 is employed in the present study, and extended to include the strain induced by the condensation of vacancies. The model assumes no ferrostatic pressure at the solidification front and no remelting. Furthermore, the model is only valid as long as the coherent part of the mushy zones does not meet at the centreline of the strand. The total strain rate is rewritten as the sum of the elastic strain rate, the thermal strain rate, and the strain rate coming from the condensation of vacancies:

tttt

vthe

∂∂

+∂∂

+∂∂

=∂∂ εεεε (14)

Here, eε is the elastic strain, thε the thermal strain and vε represents the strain induced by the condensation of vacancies. In equation 14, any volume change occurring below the coherence temperature, such as phase transformations, may be added as additional terms. The thermally induced tension is expressed:

tT

t

th

∂∂

=∂∂ αε (15)

The effect on the strain rate from the condensing vacancies is considered to be equal to the rate of change of the vacancy concentration:

tx

tv

v

∂∂

=∂∂ε (16)

22

The resulting strain in the solid shell may now be written as:

( ) ( )( )

∫ ∫

∂∂

−∂∂

−

∂∂

+∂∂

=t

t

txvv

S

e dttx

tTdx

tx

tT

txtx

S

* 0

1, ααε (17)

*t is the time at which the solidification front, Sx , passes a certain position. When considering equilibrium solidification, the rate of change of the fraction of vacancies becomes zero and the expression becomes equal to the expression derived by Kristiansson.23 Once the strain in the solidifying shell is obtained it has to be converted to the stress. Remembering that the formation of cracks occurs when the material is brittle, Hooke’s law is used to transfer strains into stresses. Hooke’s law will only give valid results within its limits of validity, which is below the yield stress. However, when the material is brittle, the yield stress becomes equal to the ultimate tensile stress.

4.4 Mathematical evaluation of the experimental work The mathematical evaluation involves solving the Fourier heat equation in one dimension with a Robin boundary condition. The boundary condition includes the heat transfer coefficient which describes the heat transport away from the strand. The heat transfer coefficient varies during solidification. During continuous casting, the heat transfer coefficient varies between the cooling zones. Figure 15 shows the cooling curves for the surface, centreline and one point midway between the centre and surface, for the two casting speeds used in the performed experiment. The heat transfer coefficient decrease in steps as the strand descends through the casting machine, which is seen on the increase of the surface temperature as the heat transfer changes. In figure 15 both equilibrium and non-equilibrium calculations are shown, but the calculated temperatures coincide and are therefore seen as one line for each of the casting speeds. It has since long been established that the recalescence of the surface due to the decreased cooling introduces higher stresses in the solidified shell which may cause cracking.25-27 The change of the stresses is illustrated in figure 16. The figure shows an equilibrium calculation for Fe-10%Ni 30 seconds after a change of the heat transfer coefficient. It is seen that the larger the decrease of the heat transfer coefficient, the larger the increase of the stress. The ultimate tensile stress, as given by the temperature profile in the shell, is also inserted in the figure, and it is seen that even a moderate decrease of the heat transfer coefficient is enough to increase the stress level above the ultimate tensile stress.

23

0 5 10 15 20 25700

800

900

1000

1100

1200

1300

1400

1500

Distance below meniscus [m]

Tem

pera

ture

[°C

]

v = 0.66 m/minv = 0.78 m/minSolidification interval

Surface

Halfway

Centreline

Figure 15. Cooling curves at surface, centreline and intermediate point for both casting speeds.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045-300

-250

-200

-150

-100

-50

0

50

100

Distance from cooling surface [m]

Cal

cula

ted

stre

ss [M

Pa]

HTC change to 900

Rupture limit

No change

HTC change to 700

HTC change to 600

HTC change to 400

HTC change to 200

Figure 16. Calculated stress distribution 30 seconds after a change of heat transfer in Fe-10%Ni.

24

Non-equilibrium solidification did not affect the temperature distribution much, but the condensation of vacancies below the coherence temperature will induce strains into the solid material. Calculations on the experiment described above are shown in figure 17, where the right side limit coincides with the centreline. The figure contain both equilibrium and non-equilibrium calculations of the stress. Inserted into the figure is also the ultimate tensile strength given by the temperature distribution in the material. The stress profiles show that non-equilibrium solidification has a stress exceeding the ultimate tensile stress close to the solidification front.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14-350

-300

-250

-200

-150

-100

-50

0

50

100

Distance from cooling surface [m]

Stre

ss [M

Pa]

Rupture limit

Equilibrium solidification

Non-equilibrium solidification

Figure 17. Calculated stress distribution for equilibrium and non-equilibrium solidification.

By gathering information of the positions where the calculated stress exceeds the measured ultimate tensile stress and the material at the same time is brittle, a crack plot can be produced. Such plots are seen for the higher casting speed in figure 18 calculated for both equilibrium and non-equilibrium solidification, where the grey surface marks cracking. For comparison the locations for the crack initiation and the widening of the cracks are marked by arrows at the bottom of the figures. The starting point of the cracks should be the position where the stress first reaches the ultimate tensile stress, and the crack will thereafter elongate into the material as long as the stress is above the critical value.

25

0 0.05 0.1

0

5

10

15

20

25

↑↑Distance from cooling surface [m]

Dis

tanc

e be

low

men

iscu

s [m

]

Two-

phas

e re

gion

Change of HTC

Change of HTC

0 0.05 0.1

0

5

10

15

20

25

↑↑Distance from cooling surface [m]

Dis

tanc

e be

low

men

iscu

s [m

]

Change of HTC

Change of HTC

Two-

phas

e re

gion

Figure 18. Calculated crack positions for equilibrium (left) and non-equilibrium (right) solidification

and casting speed 0.78 m/min.

4.5 Discussion A model to investigate the influence of thermal and vacancy induced shrinkage on the formation of cracks has been developed. The brittle behaviour of the material enables an elastic model to be used. Experimental thermomechanical data is used to predict the positions where the stress becomes critical and a crack forms. Measurements from the continuously cast strand show that the location of the cracks is moved towards the cooling surface when the casting speed is increased. The investigation of the cracks also shows that two different crack propagations occur during the solidification process. However, the two different cracking areas are not completely separated but may be distinguished by a clear widening of the cracks. Fairly good agreement is obtained for the prediction of the first crack formation, but the lack of a relaxation model prevents a better prediction of the second crack location.

26

27

Chapter 5 Macrosegregation 5.1 Introduction Macrosegregation occurs during solidification due to the redistribution of segregated liquid. The reasons for this movement of segregated liquid may be found in density differences of the material due to temperature or compositional differences as well as contraction of the material due to solidification and thermal shrinkage. In continuous casting, the centreline segregation is the most severe macrosegregation. During the final solidification of a strand, segregated liquid from above is sucked into the dendrite network in order to compensate for the shrinkage. The reason for the formation of the centreline segregation has been attributed to the thermal contraction of the already solidified shell,38,39 but also to the solidification shrinkage occurring during the final solidification.40 By applying thermal41-43 or mechanical42,44 soft reduction of the strand the formation of the opening in the centre may be prevented. Electromagnetic stirring at the end of solidification has also been reported to reduce centreline segregation.45,46 During ingot casting, the most common types of segregation in ingots are the positive and negative segregation zone and the channel segregates.47,48 The positive and negative segregation zone refers to the fact that an area with less segregated elements normally is found at the bottom of an ingot while the composition exceeds the nominal composition in the top of the ingot. The negatively segregated zone in the bottom of the ingot is explained by the sedimentation of equiaxed crystals formed in the bulk liquid,49 while the positively segregated area at the top of the ingot stems from the flow of segregated liquid towards the top of the ingot.50,51 A-segregates are channel segregates which forms due to natural convection in the mushy zone due to density variations of the melt. A liquid jet stream melts the dendrite network forming a channel where highly segregated liquid flows upwards due to its lower density. The channel is normally visible as pencil like segregates. The A-segregation is also the main reason for the positively segregated zone at the

28

top of large ingots as the channels transfer alloying elements towards the ingot top. The formation of A-segregates can be prevented by choosing the alloying elements so that the density of the liquid remains constant over the whole solidification interval.49,50,52 Maintaining the density constant will therefore also prevent the formation of the positive segregation at the top of the ingot.

5.2 Continuous casting

5.2.1 Experimental work The experimental work regarding macrosegregation in continuous casting is based on the experimental work discussed in chapter 4. Two casting speeds were used. The lower withdrawal rate was chosen so that final solidification occurred before the strand reached the area where mechanical soft reduction is performed and the higher casting velocity was chosen so that the final solidification occurred inside the area where mechanical soft reduction was applied. Samples taken from the strand were used to investigate the segregation as a function of the distance from the cooling surface. The representative segregation pattern for the lower casting speed is shown in figure 19, and the segregation pattern for a sample with final solidification inside the soft reduction area is seen in figure 20. The maximum centreline segregation was measured in all samples cut from the strand and figure 21 shows the result as function of the strand length. The casting speed was increased after 30 meters of cast strand, and was thereafter kept constant during the remainder of the experiment. In the figure, approximately the first 10 m were cast at a constant lower speed, while a transition area where the metallurgical length successively moves downward into the soft reduction zone was seen thereafter. After 30 m all samples were cast at a constant higher speed.

29

0 50 100 145 190 240 2900.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

Distance from lower surface [mm]

Segr

egat

ion

ratio

[C/C

0]

Sulphur

Carbon

Figure 19. Measured carbon and sulphur distribution in sample with final solidification without soft

reduction.

0 50 100 145 190 240 2900.9

0.95

1

1.05

1.1

1.15

1.2

1.25

Distance from lower surface [mm]

Seg

rega

tion

ratio

[C/C

0]

Carbon

Sulphur

Figure 20. Measured carbon and sulphur distribution in sample with final solidification with soft

reduction.

30

0 5 10 15 20 25 30 35 40 45 501

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Strand length [m]

Segr

egat

ion

ratio

[Cm

ax/C

0]

Sulphur

Carbon

Figure 21. Segregation ratio as function of the position in the strand.

5.2.2 Mathematical evaluation The heat transport was solved in one dimension with the Fourier heat equation. The resulting cooling curves were shown in figure 15. The calculated temperature distribution is used to calculate the shrinkage of the strand during solidification of the central parts. The contribution from the shrinkage cooling to the void forming in the centre, can be calculated as the total shrinkage of the strand thickness during the solidification of the centreline. The size of the increase of the central volume is the sum of the thermal shrinkage due to the temperature decrease:

( ) dtdxdTLS

L

t

t

x

cool ∫ ∫ ⋅=∆0

α (18)

Lt and St denote the time when the centreline reaches the liquidus and solidus temperatures respectively, x is half the thickness of the strand and α the elongation coefficient. The solidification shrinkage which may contribute to the centreline segregation is the shrinkage occurring in the central parts only. This shrinkage does not change the considered volume, but will cause a liquid flow into the volume due to the shrinkage:

31

( )∫ ⋅=∆L

Ssol dxdgL0

β (19)

β is the solidification shrinkage and L denotes the width of the examined central volume. In these calculations, the length L was considered to be equal to the width of the centreline segregation given by the chemical analysis, see figure 19. Figure 22 illustrates the shrinkage formed, and also the effect of the mechanical soft reduction. The reduction is assumed to directly affect the central void.

16 18 20 22 24 26 280

0.05

0.1

0.15

0.2

0.25

0.3

Position below meniscus [m]

Shrin

kage

dur

ing

solid

ifica

tion

of c

entre

[mm

]

Soft reduction zone

Cooling shrinkage(0.66 m/min)

Cooling shrinkage(0.78 m/min)

Solidification shrinkage(0.66 m/min)

Solidification shrinkage(0.78 m/min)

Figure 22. Calculated shrinkage of the central volume.

The central void created due to the shrinkage is filled with liquid from above. The liquid is sucked downward through the dendrite network and is therefore enriched. The thermal shrinkage adds volume to the central void while the solidification shrinkage does not contribute, see figure 23. The effect on the segregation is therefore different. The volume flow is directed towards the shrinkage volume. When the strand is compressed, the volume decreases its size and segregated liquid is squeezed out of the volume. Figure 24 shows the results from calculations on sulphur for the two casting speeds.

32

solL∆ coolL∆ Figure 23. Schematic illustration of the central volume where centreline segregation occurs.

16 18 20 22 24 26 280.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Position below meniscus [m]

Cen

trelin

e se

greg

atio

n ra

tio o

f sul

phur

[C/C

0]

Soft reduction zoneSegregation due to cooling shrinkage

(0.66 m/min)

Segregation due to solidification shrinkage(0.66 m/min)

Segregation due to cooling shrinkage(0.78 m/min)

Segregation due to solidification shrinkage(0.78 m/min)

Figure 24. Calculated centreline segregation of sulphur due to thermal and solidification shrinkage.

33

5.3 Ingot casting

5.3.1 Experimental work The macrosegregation in one octagonal 12 ton ingot and one rectangular 10 ton ingot has been investigated. The ingots were cast by uphill casting and the target composition was identical to that used in the continuous casting experiment. The cooled ingots were cut in pieces in order to enable sulphur printing and drilling for testing of chemical composition. Sample drilling was performed over the vertical cross section in order to obtain the segregation pattern of C and S and the chemical composition for the rectangular ingot is shown in figure 25. In the figure, the centreline composition is shown to the left and the concentrations along three vertical lines from surface to centre are shown to the right. The arrows in the picture mark the locations of the horizontal lines. A sulphur print of the corresponding surface is shown in figure 26. In the rectangular ingot, an increase of the segregation ratio is seen towards the top of the ingot. The horizontal lines shown in figure 25 show that the composition is rather even towards the surfaces at all levels. At higher levels the ingot has a clearly unstable segregation ratio towards the centre of the ingots. The instability of the horizontal lines toward the centre coincides with the appearance of A-segregates, seen in the sulphur print in figure 26. At the bottom, the ingot shows a somewhat increasing negative segregation toward the centre.

0.5 1 1.5 2 2.50

200

400

600

800

1000

1200

C/C0

Dis

tanc

e fro

m b

otto

m o

f ing

ot [m

m]

←

←

←

CS

0 100 200 300 4000.5

1

1.5

2

2.5

C/C

0

0 100 200 300 4000.5

1

1.5

C/C

0

0 100 200 300 4000.5

1

1.5

Distance from centreline [mm]

C/C

0

Figure 25. Carbon and sulphur composition in rectangular ingot.

34

Figure 26. Sulphur print of vertical cross-section of rectangular ingot.

35

5.3.2 Mathematical evaluation A theoretical evaluation of the entire segregation process during solidification of large ingots is quite complex. Several advanced calculations of macrosegregation involving continuum formulations have been presented in the literature,52-55 showing reasonable results. These codes are, in a general case, able to predict the formation of A-segregates.52,55 However, when the ingot size is large, the mesh size makes it difficult to resolve the channel segregation in this type of simulations. The calculations which are performed here, utilises a simpler approach to calculate the influence of the A-segregation on the total macrosegregation in large ingots. During solidification, the density of the liquid metal changes due to the segregation of alloying elements. A decrease of the density causes the liquid to move upward. The effect of temperature and segregation on density was investigated by Olsson,56 and his result is used in the calculations together with the lever rule and Scheil’s equation in order to describe the density change of the liquid, and the result is shown in figure 27. It is to be noted that the segregation of the heavy elements, such as Mo, increases the density while the segregation of lighter elements decrease the density. A carefully chosen initial composition could therefore equalize the effect from the density decreasing element with those elements which increase the density in order to decrease the density change during solidification and thereby decrease the macrosegregation.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-300

-250

-200

-150

-100

-50

0

50

Fraction solid [-]

Den

sity

cha

nge

[kg/

m3 ]

Figure 27. Calculated change of liquid density during solidification.

36

The compositional changes due to sedimentation are calculated according to equations 20 and 21.49 The concentration in the sediment zone, sed

iC , and the concentration in the remaining liquid, L

iC , is calculated assuming that each part of settled solid is followed by W parts of liquid and a relative height of the sediment zone fV . The relative height of the sediment zones was established from the sulphur prints.

( )( ) fi

iisedi VkW

WkCC−−++

=11

0

(20)

( )W

VkCC

fi

iLi

+

−−

=

11

1

0

(21)

0iC is the initial composition and ik is the partition coefficient of element i.

The effect of the A-segregates can be expressed as a mass balance considering the concentration of an element in the bulk liquid and the concentration of the same element in the liquid in the A-segregate as expressed in equation 22.49

( )Li

AiAA

Li CCAvV

dtdC

−= (22)

The volume flow of the segregated liquid in the A-channels is given by the velocity, Av , and the total cross sectional area of the A-segregates, AA . The concentration of the bulk liquid, L

iC , is initially given by equation 21. By comparing a chemical analysis of the A-segregate, A

iC , with Scheil’s equation the fraction solid was determined to be 0.7 when the A-segregates form, which is also consistent with observations made by Suzuki and Miyamoto.57 The liquid velocity is derived from Darcy’s law.58 The total cross sectional area of the A-segregates will vary with time and is found by relating the A-segregate density to the location of the solidification front. The density of A-segregates is estimated from the horizontal sulphur prints. Heat conduction simulations were performed using a commercially available software, NovaFlow & Solid™, in order to obtain the location of the solidification front as a function of time. Equation 22 may be solved to reveal the segregation resulting from the flow of segregated liquid in the A-channels. The result is plotted along with the measured data for the rectangular ingot in figure 28.

37

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.40

200

400

600

800

1000

1200

1400

1600

Segregation ratio [C/C0]

Hei

ght [

mm

]

Sulphur (measured)

Carbon (calculated)

Sulphur (calculated)

Carbon (measured)

Figure 28. Calculated and measured composition along the centreline in the rectangular ingot.

5.4 Discussion Ingot casting and continuous casting experiments has been performed with a steel of the same grade. Macrosegregation occurs in both experiments, but the origin of the segregation is essentially different. In the continuous casting experiment, centreline segregation is seen. The centreline segregation is caused by a transport of liquid downwards along the centreline during the solidification of the centre of the strand. This is an effect of the solidification and cooling shrinkage, which acts to part the strand into two halves by forming a crack in the centre. When mechanical soft reduction is applied the segregation is prevented. The structure and macrosegregation in the two ingots cast in this study is similar to the generally reported segregation pattern,47,48 a negatively segregated zone in the lower parts of the ingots and an increasing composition towards the top. The primary source of the macrosegregation in the ingot cast material is the transport of segregated liquid in the A-segregates.

38

39

Chapter 6 Concluding remarks Modelling of shrinkage during solidification of aluminium and iron based alloys have been performed along with experimental work showing that the effect of vacancy condensation has to be included in modelling work of solidification processes. Air gap formation was studied in a cylindrical mould. Displacements and temperatures were recorded during the solidification process. Consequent theoretical modelling was done where the effect of non-equilibrium solidification was included. In iron based alloys cast in a water-cooled copper mould the vacancy condensation proved to be less significant due to the large temperature gradients, while the condensation of vacancies contributed largely to the shrinkage in the solidification of aluminium based alloys where the cast iron mould gave smaller temperature gradients. The agreement between experimentally determined displacement and theoretically calculated shrinkage was good. Crack formation during solidification was investigated. A one-dimensional model to predict the crack locations was developed and applied on a continuously cast slab, where crack positions were experimentally determined. Calculations were performed for both equilibrium and non-equilibrium solidification and the results showed clearly that vacancy condensation has to be included in order to properly predict crack formation. The shrinkage of a continuously cast strand also causes macrosegregation to occur along the centre of the strand, known as centreline segregation. The contribution to the centreline segregation by the cooling of the solidified material was compared to the contribution by the solidification shrinkage, and it was theoretically found that the cooling shrinkage is the primary cause of the centreline segregation. Finally, the macrosegregation in the continuously cast steel was compared to the macrosegregation seen in ingot casting which is caused primarily by density differences in the liquid.

40

41

Chapter 7 Future work This thesis has dealt with the shrinkage during the solidification process and some of the resulting effects. Experimental measurements have been compared to numerical results that clearly show that the condensation of vacancies does influence the shrinkage and subsequently the solidification phenomena that are caused by density changes in the solid state. Detailed information on how the incorporation of condensation of the vacancies is an area in which further investigation is needed. The amount of excess vacancies incorporated into the solid is one such question. A more detailed knowledge of the behaviour of the vacancies in the mushy zone is another. The models presented to calculate the shrinkage in the shell are both one-dimensional although with different coordinate systems. In order to be able to predict the air gap formation or cracking in systems with more complex geometries the gained knowledge on the shrinkage behaviour has to be extended to more generalised formulas. The model for prediction of cracks may be further extended. The model does not consider relaxation when cracks are predicted which makes the further prediction difficult after the point where the ultimate tensile strength has been reached. Although the simplicity of the model depends on the use of purely elastic models, another step would be to extend the stress model to regard also creep and plasticity. When dealing with numerical calculations, the data inserted into the calculation is important. Input data includes all traditional material properties which have to be known for the material studied. Considering the extension of the shrinkage theory to include the condensation of vacancies, new information is needed in order to more precisely describe the non-equilibrium solidification process.

42

43

Acknowledgements During my work with this thesis, I have received a lot of help from different people and organisations, in the shape of guidance, financing, discussions, material, general support and plain cheering. I would like to express my deepest gratitude to all of you: Prof. Hasse Fredriksson for supervising me throughout this work. The Swedish Energy Agency, the Swedish Steel Producers’ Association, the Brinell Centre and SSAB Oxelösund AB for financial support of the different parts of this work. SSAB Oxelösund AB and Scana Steel Björneborg AB for support with equipment and personnel during the experiments. The members of the Swedish Steel Producers’ Association committees JK24045, JK24047 and JK24048 for helpful and interesting discussions. The members of the committee for comparative studies of ingot cast and continuously cast tool steel. Christer Offerman, Per Hansson, Jan Sarnet and Björn Widell deserve special mentioning. The former and present staff at the Casting of metals. I am especially indebted to Anders Eliasson for always having time for advices and discussions about experimental work. My dear friends! Without all my friends there would be a lot less of the relaxing good times. All of them can’t be mentioned here, but I would like to point out a few names: Magnus Bergstrand, Jonas Bergstrand, Mikko Östlund and Jessica Elfsberg. The Lagerstedt family, especially mom and dad for endless love and support My extended families: the Helmrich family, the Kron family and the Paral family. Finally, Jenny Kron for everything.

44

45

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