21.

21.
Dilatometric investigation on the phase
transformations during thermal treatment of
a low silicon TRIP-aided multiphase steel
M
l i
I
S
I
TUDelft
UCL
Delft
U n i v e r s t t v
of
Tecnn
Université
catholique
de Louvain
This work is submitted as
part fulfilment for the
Degree of Materials
Science Engineer.
Supervisors :
Prof. F. Delannay
Prof. S. van der Zwaag
Valéry ROLIN
Academic Year '99 - '00
Acknowledgements
In the chronological order of the elaboration of this thesis work,
I would like to thank Professor Francis Delannay for having helped me to go and
study at the TUDelft
I thank especially Dr Lie Zhao for his interest in my work and the time he spent with
me.
I also thank Professor Sybrand van der Zwaag, Dr Jilt Sietsma, Theo Kop, Yvonne
van Leeuwen and Pieter van der Wolk for their help, their judicious advices and their
sympathy.
I still would like to thank Pascal Jacques for his 'internet' pieces of advice.
And I owe a special acknowledgement
to Anne Mortens for all the time she has
given to me, and for her conscientious help.
2
Abstract
TRIP-aided multiphase steels owe their high strength and high formability mostly to the
austenite that they contain at room temperature. This is what makes of them the challenging
material for automotive industry. In the low-alloyed steels, the retention of the austenite is
obtained by a specific thermal treatment, which includes a bainitic transformation. In order to
prevent the precipitation of cementite in the enriched austenite, a graphitising element such as
silicon is usually added to the composition. But the presence of silicon creates several
problems within the finished products, this is why researches are carried out on low-silicon
TRIP-aided multiphase steels.
The main experimental technique that was used is dilatometry. (1) In a first time the allotropic
transformation a ^ y upon heating was studied, A Matlab program was designed for the
calculation (on the basis of the experimental data) of the amount of austenite that appears in a
sample during an intercritical annealing, (2) In a second time, several thermal treatments have
been designed in order to get information for the drawing of TTT diagrams. Therefore,
samples were quenched from the austenitic region to several temperatures where an
isothermal holding was applied for 15 minutes. Once again, the measured data was used by a
Matlab program for the calculation of the fraction transformed, which finally lead to the TTT
diagram.
Scanning electron microscopy as well as optical microscopy were used on the samples in
order to verify the results found by dilatometry and calculation. (1) Image analysis allowed to
measure the distribution of phases determined by the regulation of the intercritical annealings.
(2) And, by comparing the samples having undergone the different isothermal holding, it was
possible to determine the influence of the heat treatments on the microstructures.
3
Table of contents
1.
INTRODUCTION
6
1.1
Objectives
6
1.2
TRIP-aided multiphase steels
7
1.2.1
1.2.2
1.2.3
1.2.3.1
1.2.3.2
1.2.4
2.
TRIP effect
Usual processes for the production of TRIP steels
Stabilisation of austenite during the bainitic transformation
Bainitic transformation
The incomplete reacüon phenomenon
Low silicon TRIP-aided multiphase steels
M A T E R I A L AND M E T H O D S
2.1
2.2
2.2.1
2.2.2
2.3
3.
13
Material
13
Characterisation
Dilatometry
Microstructural characterisation
16
16
18
Calculations
2.3.1
Dilatation-Phase
2.3.1.1
Lever rule
2.3.1.2
Calculation based on the lattice parameters
2.3.2
Newton-Raphson
2.3.3
Matlab
fraction
RESULTS
3.1
3.1.1
3.1.2
3.1.3
3.1.4
3.2
20
20
20
21
24
25
26
First measurements
Characteristic temperatures
Expansion coefficients
Transformation curves
MTData calculation of the equilibrium
26
26
28
29
33
Calculation upon the heating data
3.2.1
Calculation of the expansion coefficient of the ferrite
3.2.2
Lattice parameter of the ferrite and the cementite
3.2.3
Transformation curve
3.2.3.1
Calculadon
3.2.3.2
Microstructure analysis
3.2.4
Fitting parameters
3.3
7
8
9
9
10
11
34
35
35
37
37
43
45
3.3.1
3.3.2
3.3.3
Bainitic transformation
Isothermal holdings
Microstructure analysis
Calculation on the bainidc holding
46
46
52
55
3.4.1
3.4.2
3.4.3
Intercritical annealing at 750°C : test
Introduction
Dilatometry
Microstructure
57
57
59
61
3.4
4
4.
DISCUSSION
4.1
Heating calculation
4.2
Bainitic transformation
4.2.1
Synthesis of the results on the isothermal holdings
4.2.2
Quenches
4.2.3
CCT Diagram
4.2.4
Stability ofthe calculadons
4.3
T T T diagram for the annealing at TSO^C
4.4
Decayed series
Quenches from 900°C
Elements for CCT diagrams
4.4.1
4.4.2
4.5
Remedy: Formula applied upon the quench
4.6
Intercritical annealing at 750°C : test
Explanations for the gap between the annealings at 750°C
Microstructure
4.6.1
4.6.2
4.7
Amount of retained austenite
5.
CONCLUSIONS
6.
APPENDIX
6.1
GuidelinesManual bainitic transformation analysis program
6.2
Programs
6.3
Three-dimensions T T T diagram
1. Introduction
Objectives
1. Introduction
1.1
Objectives
As suggested in its title, this work is an investigation on the thermal treatments for the
production of a low silicon TRIP-aided multiphase steel. This material belongs to the family
of high-strength foimable steels. The exceptional mechanical properties of the TRIP-aided
multiphase steels have been studied for several years by now, and they will be detailed in this
introduction. What is more recent is the interest bom to the low silicon family, which should
be more appropriated on a technological point of view for industrial production.
Practically, a low carbon - low silicon steel was chosen to be studied in a first time by means
of dilatometry, and eventually by other chai-acterisation techniques : optical microscopy,
scanning electron microscopy and X-ray diffraction.
Besides the experimentation, the second main part of the work has been the development of a
calculation method for the interpretation of the dilatometric data. Two kinds of
transfomiations were analysed by this method: (1) the allotropic transformation a-^y upon
heating, i.e. the calculation of the fraction of austenite that appears during an intercritical
annealing, (2) the isothermal transformation in the bainitic region following a quench from
the austenitic region.
The idea of these calculation was to get data on the fraction transformed for the drawing of
TTT diagrams corresponding to three annealing temperatures : two intercritical, and one over
Ac3.
In the whole set of dilatometric experiments, some have served for an additional investigation
on intercritical annealings. The obtained microstructure still constitutes a mystery since it
cannot be explained from the thermal treatments undergone by the samples.
6
1. Introduction
TRIP-aided multipl^ase steels
1.2 TRIP-aided multiphase steels
Steel is still the first choice material for the construction of car bodies in the automotive
industry. In the current conjuncture, the big challenge is the fuel consumption, which
demands a reduction of the weight. I f the iron and steel industries want their material to keep
its place, they must improve its strength. This can be achieved with the TRIP effect
.
Although they are low alloyed material, the TRIP-aided multiphase steels show an excellent
combination of strength and ductility. These performances can be attributed to the
TRansformation Induced Plasticity, i.e. the continuous transformation of retained austenite
into maitensite during straining ^^'^^.
1.2.1 T R I P effect
The schematical free energy curves on Figure 1 help to understand how the mechanically
induced transformation of the austenite works. For a fixed composition. To is the equilibrium
temperature between austenite and martensite, and Ms is the temperature at which the
undercooling is sufficient to provoke transformation. I f Tj is an intermediate temperature
between Ms and To, the austenite that exists at Ti is said to be metastable since its free energy
curve is above the curve of the martensite. The fact is that the difference between the free
energy curves at T, (AG^^"')
has not yet reached the critical driving force (AG^^^"').
However, it is possible for the austenite at Ti to transfomi into martensite i f a sufficient
mechanical energy is provided (U'). This mechanically induced transformation bears the name
of TRIP effect.
M.
„.^^^S. Martensite
s
C3
O
Austenite
Ms
Tl
To
Temperature
Figure I: Free energy curves of tiie austenite and tiie martensite
7
1. Introduction
TRIP-aided multiphase steels
1.2.2 Usual processes for the production of T R I P steels
TRIP-aided multiphase steels can be produced in two ways : (i) heat treatment after cold
rolling
(i)
and, (ii) a continuous thermomechanical treatment including a hot rolling
.
Figure 2 shows the heat treatment used for cold-rolled TRIP-assisted multiphase
steels. The aim of this process is to stabilise some austenite at room temperature thanks to a
two steps carbon enrichment. In a first time, the austenite is formed during the intercritical
annealing and it is enriched in carbon coming from the neighbouring ferrite
. Secondly, a
bainitic holding provokes a partial transformation ofthe intercritical austenite into bainite, and
the carbon enrichment of the remaining austenite, which is therefore not transformed into
martensite during the fmal quench to room temperature
. Those low alloyed steels usually
contain up to 1.5 - 2.0 wt. % silicon in order to prevent the precipitation of cementite in the
carbon-enriched austenite. During the further processing of the material, the metastable
austenite can transform into martensite when there is a strong plastic deformation.
(ii)
The processing of TRIP steels via thermomechanical treatment is quite similar, except
that the material is hot-rolled, then directly cooled to the intercritical region, where the
annealing is carried out in a continuous way. The continuation ofthe process is identical to
the one for cold-rolled steels (i).
Figure 2: Typical heat treatment scheme for the production of TRIP-aided multiphase steel
1. Introduction
TRIP-aided multipimse steels
1.2.3 Stabilisation of austenite during the bainitic transformation
1.2.3.1 Bainitic transformation
Before entering into the specificities of the multiphase TRIP-aided steels, it is necessary to
examine the bainitic transformation. It is worth remembering that the bainite is the
microstructure that forms when a steel is quenched from the austenitic region to a range of
temperatures below the femtic region and above the martensitic region. While the formation
of ferrite is a diffusive transformation, the formation of martensite is displacive. What about
bainite ? It has been accepted now that its formation is displacive, even though there is some
diffusion in the nucleation process
.
Austenite
Grain Boundary
Figure 3: Sciiematic representation of tiie development ofa sheaf of bainite
The growth of the bainite is sketched in Figure 3. The sub-units have the cristallographic
lattice of the ferrite, and are sometimes called "bainitic ferrite". They nucleate at the grain
boundaries of the austenite and grow very fast as a plate until the dislocation pile-up at the
austenite/ferrite interface stops them. Then new nucleation occurs at the tip of the laths,
leading to a sheaf stmcture.
As their fonnation is displacive, the fenite laths are supersaturated with carbon during a short
instant, after their formation, carbon then diffuses into the surrounding austenite. This
becomes the place for caitide precipitation, unless silicon or aluminium has been added.
These elements, which ai'e substitutional, do not diffuse during bainitic transformation. Two
diffusion processes exist that lead to two kinds of bainite, as sketched in Figure 4.
9
1. Introduction
TRIP-aided multipliase steels
Formation o f a supersaturated Sub-Unit
Cai-bon diffusion
into austenite
/
2
\
Carbon diffusion into
austenite and carbide
precipitation i n feixite
C
3
Carbide precipitation
f r o m austenite
Upper Bainite
1
Lower Baiiiite
Figure 4: Scliematic illustration of tiie formation of either upper bainite or lower bainite
The difference between upper bainite and lower bainite comes from kinetic effects
Upper
bainite appears when the transformation is performed at such a high temperature that the
carbon diffusion out of the ferritic laths can take place very quickly. On the other hand, at
lower temperatures, a part of the carbon cannot diffuse out of the lath and thus, carbides
precipitate inside. This is the case of the lower bainite.
1.2.3.2 T h e incomplete reaction p h e n o m e n o n
The fact that some austenite can be stabilised during the formation of bainite is called an
incomplete reaction ^^1 It will be showed hereafter that this incomplete reaction is possible
because the bainitic transformation is displacive.
Figure 5 presents the free energy curves of ferrite and bainite at the temperature T,
represented as a function of the carbon content. The crossing of these curves defines the
carbon concentration above which the austenite cannot transform into ferrite in a displacive
way. The line that represents this maximum carbon content as a function of the temperature of
transformation is called the To curve. The result is that, for an isothermal transformation at Ti,
and i f there is no carbide precipitation in the austenite, the bainitic transformation will stop
when the caiton concentration in the austenite reaches the value defined by the To curve.
10
1. Introduction
TRIP-aided multipliase steels
On the other hand, i f the austenite had transformed by a reconstructive process, the carbon
concentration in the ferrite and in the austenite would have been respectively defined by the
Aei and the Aea lines, according to the tangent mle.
Carbon Concentration
Figure 5: Scliematic illustration of tiie origin ofthe To cwve on a piiase diagram as resulting
at each temperature of specific points of the free energy curves offerrite and austenite
Besides the fact that the bainitic transformation must be displacive, a second condition for the
retention of austenite is that there must not be any carbide precipitation. It is possible to get
rid of that precipitation by adding silicon to the steel. Indeed, the solubility of the silicon in
the cementite is very small and therefore carbon cannot precipitate ^''l I f this requirement is
met, the high carbon content of the austenite has the effect to drop Ms below room
temperature, and the austenite can therefore be retained.
1.2.4 Low silicon TRIP-aided multiphase steels
Significant amount of austenite can be retained in bainitically transformed steels highly
alloyed with a silicon concentration around 2 %
and it is accepted that a conventional
TRIP-aided multiphase steel must contain at least 1 wt. % of silicon ^ " l But the problem is
that besides its good effect on the hindering of the cementite precipitation, silicon, when it is
added to more than 0.5 wt. % in steel, creates problems on the finished product. Firstly, a
silicon oxide layer appears on the material after the hot-rolling. Secondly, at the galvanisation,
too much silicon provoke the formation of intermetallics Fe-Zn that renders the protection
11
1. Introduction
layer fragile
TRIP-aided multipiiase steels
\ This is why a real interest exists for low silicon TRIP-aided multiphase
steels. The steel that is going to be studied in this work has the following composition : 0.16
wt. % C, 1.5 wt. % Mn, and 0.4 wt. % Si, which is similar to the composition of typical coldrolled dual-phase steels '^^^l As a matter of fact, the silicon content is much lower than in
conventional TRIP-aided multiphase steels, and it should be difficult to get retained austenite.
However, the dilatometric-investigation is interesting since it can bring many infoimation on
that grade of steel.
12
2. Material, and metixods
Material
2. Material and methods
2.1
Material
The one and only material studied in this work is a low silicon steel that was provided by the
research and development department of Hoogovens
As it was to be studied by
dilatometry, we have received the samples of industrially produced hot-rolled FeCMnSi steel
in the shape of solid cyhnders of 10 mm long and 4 mm in diameter.
Wt. %
Mn
Si
C
1.5
0.4
0.16
Table 1: Composition given by Hoogovens.
This steel was said to have a composition with lean chemistries, and that means a maximum
of 0.16 wt. % C, 1.5 wt. % Mn and 0.4 wt. % Si. Those values were retained for the
calculations based on the lattice parameters. Meanwhile, in order to have a second source for
the composition, we confided three samples, as they were received from Hoogovens, to the
department of Chemical Technology of Delft.
The samples were digested in a mixture of 20 ml aqua regia and 5 ml HF in a closed Teflon
vessel using a microwave furnace. The ICP-OES technique was used to analyse the sample
solutions. The concentrations found are :
Weight [mg]
975.5 ± 0 . 8
Wt. %
Mn
Si
C
1.47 ±0.015
0.27 ±0.015
0.14 ± 0
Table 2: Composition measured at tiie TUDelft.
The silicon concentration is much lower than the previous value of 0.4 wt. %. That would
mean that this is a "very low silicon steel". Anyway, we will see that it will not be a problem
for the calculations as the silicon concentration does not play any role. An explanation that
was proposed for this lower measured value was that a precipitate including a part of the
silicon forms in the batch, so that this part is not blown in the plasma torch.
On the other hand, the measured concentration for the carbon was also lower than the value
given by Hoogovens (0.14 < 0.16). This has more critic consequences as it influences the
results of the calculation. Nevertheless, we made the choice to work with the first numbers.
We asked our contacts in Hoogovens to repeat the same composition measurements with
more details (Table 3). The results follow in weight percent:
13
2. Material and methods
Wt. %
Material
Mn
Si
C
Al
Ni
Cr
Cu
P
S
Mo, Sn
1.48
0.414
0.154
0.046
0.022
0.019
0.015
0.012
.0.011
<0.001
Tahle 3: Composition measured by Hoogovens.
These measurements confirm the choice to take into account the three first alloying elements
(Mn,
Si, C), and to assume that their concentrations are 1.5, 0.4 and 0.16 wt. % réspectively.
Internet provides some information on steels: the web site of the university of Cambridge,
department of H.K.D.H.Bhadeshia, can release TTT
and CCT diagrams '^^"l
1200
•TTT
•CCT
1100
1000
P
900
|-
800
700
0.16*C, 0.4»Si, 1.5»In,
m\i
OSSMo, Oürcr, OSfT, OjfCo, 0 rt.ppm E
Austenitizing Temperature = U73K
Cooling Rate Range = 0.01 <-> ISOK/s
600
500"
0.01
I
I 11
ml
oml
L
0.1
10
100
1000
Time (s)
Figure 6: TTT and CCT diagrams calculated on Cambridge web site ^'''l
The chart on Figure 6 assumes that the material has been austenitised at 900°C.
Pictures 1 and 2 show SEM micrographs of the material as it was received from Hoogovens.
At a low magnification, picture 1 shows the strongly banded stmcture oriented in the rolling
direction. This is due to the microsegregation that takes place during the solidification of the
liquid metal. Dendrites of ferrite 6 form in an oriented way and the manganese remains
preferably in the liquid phase. The banded stmcture is created by the mechanical deformation
at high temperature of those dendrites. The presence of pearlite (seen on picture 2) is strongly
dependent on the local concentration of manganese since this element has the effect to move
the eutectoïd point to the left. Moreover, as the manganese lessens the eutectoïd level ^^^^i.e.
the A l temperature), its distribution has an effect on the microstructures that form upon
cooling or heating through the (a,y) phase transfonnation. When cooling, the first ferritic
grains will appear in the regions poor in manganese. When heating, the first austenitic grains
14
2. Material and metiiods
Material
will appear in the regions rich in manganese. The pictures 1 and 2 allow to think that the
average grain size is around 5 pm.
Picture 1 (AR): SEM micrograplr siiowing tlie hot-rolled microstructure ofthe
received from Hoogovens. A strongly banded structure is visible.
material
Picture 2 (AR) : SEM micrograph showing the ferritic-pearlitic microstructure of tlie material
received from Hoogovens. The grain size is 5-10 jlm.
15
2. Material and metiiods
2.2
CItaracterisation
Characterisation
2.2.1 Dilatometry
The main technique used in this investigation is dilatometry. A l l the experiments have been
caiTied out in the Materiaalkunde at the Technische Universiteid Delft on a BAHR805. This
appai-atus belongs to the new generation of the technical instruments. Figure 7 might help to
understand how the dilatometer works.
Thermocouples
LVDT
Sample
Induction coil
Quartz rods
Figure 7: Scliematic illustration ofa
dilatometer.
The sample, a solid cylinder 10 mm long and 4 mm in diameter is wedged between two quartz
rod tensed by a spring. Two coils surround the sample ; one is for the heating by high
frequency induction and the other one projects a cooling gas during the possible quenches.
Two cooling gases are available ; nitrogen and helium. Nitrogen was used during the first
experiments that required a quench, but it appeared cleariy that it was not efficient enough.
Thus helium was chosen, because of its higher calorific capacity, although it is more
expensive.
Two thermocouples are welded on the surface of the sample. Only the temperature measured
by the first one (welded in the middle of the sample) will be taken into account for the
application of the temperature program, however the data measured by the second
thermocouple can be used for the calculation of the temperature gradient that exists between
the middle and the tips of the sample. The dilatation measurement employs a device called
"Linear Variable Displacement Transducer". The value of Al(T) is set to zero at the beginning
of the program, and this is always at the room temperature.
16
2. Material and metliods
Characterisation
Experiment
Quenching mode
Bainitic holding
Temperature range
Gauging experiment
No
No
, 20-900 °C
For image analysis
Yes
No
20-800 °C
Isothermal trans.
Yes
Yes
20-900 °C
Test: 750 °C
Yes
Yes
20-950 °C
Tahle 4: Set of dilatometric experiments peifonned in tiiis tiiesis work.
In order to have an idea of the range of temperatures covered by the experiments, one may
look at the Table 4. About 55 experiments have been carried out. The two first ones were
"gauging experiments", i.e. the aim was to measure the characteristic temperatures of the steel
as well as its expansion coefficients. It is just a slow heating to the austenite region, followed
by a slow cooling. A few experiments called here "For image analysis" consisted in an
annealing, intercritical or not, followed by a quench to room temperature.
Most of the work was actually spent on the "Isothermal transformations" group; about twenty
five experiments have been carried out according to the following process :
The sample is heated at the rate of 100 K/minute to one among three annealing temperatures :
750°C, 800°C or 900°C. It is annealed for 10 minutes, then quenched with hehum gas to a
level between 100°C and 500°C by steps of 50°C. This level is maintained for 15 minutes and
is followed by a free cooling to the room temperature at the rate of 100 K/min. Those
treatments are illustrated on Figure 8.
Treatments
Time [mini
Figure 8: Representation ofthe dilatometric experiments peiformedfor
isothermal transformations.
17
tiie analyse of
2. Material and methods
Characterisation
The horizontal dotted lines define A c l (= 725°C) and Ac3 (= 850°C). Finally, a few more
"exotic" experiments, which were called "Test at 15Q°C\ were performed, A maximum
temperature of 950°C is reached on the whole set.
One must be aware that the BAHR805 is not just a conventional dilatometer. Indeed, a
conventional device is just able to measure a change in length while the temperature is
evolving. It is thus adapted for gauging measurements : slow heating, slow cooling. On the
other hand, the apparatus used in this work is a "quench" dilatometer. It means that one can
include quenching modes in the program, and the big interest is to measure the dilatation that
takes place during an isothermal holding that follows the quench. This option is essential for
someone who wants to draw TTT diagrams.
It is very important to check that the sample did not move between the rods during the
experiment because it would introduce a jump in the change in length data. Unfortunately, it
often happens that the gas blown destabilises the sample during the quench, ant it renders the
experiment decayed.
When
performing dilatometry experiments, one should always be
aware
of the
decarburisation problems. At high temperatures, some carbon close to the surface may leave
the sample because of the strong diffusion. If so, the following measurements and calculations
would be falsified since the composition would have been locally changed. This is why it is
important to look at the surface of a slice cut in the sample in order to see i f there is, or not, a
segregation between the edge and the centre. Concerning this work, a checking was done on
some samples austenitised during 10 minutes at 950°C and no decarburisation was observed.
Indeed, this phenomenon can appear only from higher temperatures (at least 1100 °C).
2.2.2 Microstructural characterisation
Some of the samples analysed by dilatometry were chosen for microstructural examination.
Two slices about 1 mm thick were cut in each of them : one was taken in Louvain-ia-Neuve
for microscopical observation, and the other one was kept by Dr. Lie Zhao, from the
Materiaalkunde, TUDelft, for X-ray analysis. Those samples were cut very carefully with a
diamond saw so as to avoid the mechanically induced transformation of retained austenite.
(TRIP effect).
Microstructures were studied by scanning electron microscopy (SEM) and by optical
microscopy. The polishing has been done to a fineness in the diamond powder of 0.25 p.m. In
order to make possible the distinction between retained austenite and martensite by SEM
observation, specimens were first annealed for 2 hours at 200°C and then etched with 2 %
18
Characterisation
2. Matenal and methods
nital (Norvanol + 2 % HNO3). Because of the carbide precipitation during the annealing, the
maitensite appears as finely cracked grains, while austenite grains remain perfectly smooth
[16]
The image analysis was performed on pictures scanned from the SEM with a semi-automatic
routine working with Visilog. The application was the measurement of the fraction of phase in
quenched samples. It required fifty pictures magnified around 1400 times and taken' all around
the facet of the sample. For each picture, a brightness threshold has to be defined in order to
distinguish the ferrite from the martensite. It is important to take pictures properly scattered
on the entire available surface because there may be some variation of the phase's fraction
depending on the situation with respect to the edge of the sample.
19
Calculations
2. Material and metiwds
2.3
Calculations
2.3.1 Dilatation-Phase fraction
As for the majority of the materials, steel undergoes a dilatation when it is submitted to a rise
in temperature. I f there is no phase transformation during the heating, the dilatation can be
described by mean of an expansion coefficient. If there is a transformation, one must take into
account the different crystallographic phases that coexist at a given temperature.
Two conventional techniques for the calculation of the phase fractions, for instance during a
cooling from the austenitic region to the ferritic region, ai"e presented here. The first one bears
the name of "lever rule", whereas the second method involves the lattice parameters. (This
lever rule should not be mistaken with the lever rule relevant for the phase diagrams.) The
second method constitutes the basis of all further calculations in this work.
2.3.1.1 L e v e r rule
The volume fraction of austenite can be calculated as follows:
fer-aust
The lever rule method is illustrated on Figure 9 and by equation (1). The application of this
method requires the assumption of proportionality between the phase fraction and the length
change. By virtue of extrapolation of the linear expansion and contraction curves of the ferrite
phase and the austenite phase, we can define these values :
• far = Length change of a heated sample that would remain 100 % ferritic.
• aust = Length change of a cooled sample that would remain 100 % austenitic.
• cil (change in length) = Actual change in length of a sample cooled from 900 °C at 3K/min.
20
2. Material and metiiods
Calculations
Lever rule
110
30
500
550
600
650
700
750
Temperature P d
800
Figure 9: Illustration of tiie 'lever rule' on a dilatometric cooling curve.
Those values are temperature dependent, and, in this example, it is at 700°C. The lever rule is
very easy to apply, but the assumption of proportionality is too strong. Furthermore, using this
technique could not help to measure the amount of a new phase growing during the isothermal
level of an intercritical annealing, or of a bainitic holding. As a matter of fact, precisely that
kind of calculation was to be performed in this thesis work. Another kind of calculation has
thus to be used.
2.3.1.2 C a l c u l a t i o n b a s e d on the lattice parameters
The following developments mainly come from an article of Lie Zhao ^ ^ ' l As explained in the
previous pages, the dilatometer allows to measure the dilatation of a sample, which follows a
temperature program, previously defined by the user. But it is technically easier to measure a
length change
(AI
= 1-IQ)
than a volume change
(AV
= V-VQ),
and those values can be
linked thanks to the following formula.
1+
AV
(2)
0
Al
0
1 AV
(3)
3 V0
Another way to write the equation (3) is:
21
2. Material and methods
Calculations
Where Al - l(T) - k . The values k and Vo characterise the initial dimensions of the sample.
These values are going to be the reference in comparison with which the variation is
calculated. The initial dimensions are defined as the dimensions of the sample at the
beginning of the experiment, or at the beginning of the isothermal transformation. V'(T) is the
instantaneous volume and it is a function of the temperature.
Let us consider now the case of a phase transformation taking place upon cooling. We
suppose that there is only one product phase and one parent phase involved, for instance, the
ferrite (a) phase and the austenite (y) phase. We have to consider now the volume fractions
and fy. The instantaneous volume V'(T) of the equation (4) can be expressed as follows :
The sum of the volume fractions of the phases remains very logically equal to 1. Equations (4)
and (5) establish the relation between the volume fraction and the length change. The next
step will be to decompose the terms Va and Vy by means of the lattice parameters and the
expansion coefficients multiplied by the temperature. This will introduce the link between the
change in length and the temperature.
Here is the application of this type of calculation to a transformation that occurs during an
isothermal bainitic holding. Some values like the lattice parameters and the expansion
coefficients are still unknown but the calculations that will be made on the heating curve of
the gauging experiments will give them. Anyway, those values will be used in a formula that
figures out the fraction of retained austenite all along the isothermal holding. Two different
formulas must be written : one relevant for the completely austenitised samples, and the other
one for the cases when the quench is carried out from an intercritical region.
Austenitised samples :
A/ _ (2 * fa-B
^0
* « a - B + fy-B
* ^J-B)
'
«r-A
(6)
3*4_.
The sign '-A ' means "in the austenitising region" and '-B ' stands for "bainitic". As a matter of
fact, it is considered in the following calculations that the transformation that occurs during
the isothemial holding produces bainite, although we are aware that it is not pertinent in the
22
2. Material and metiiods
Calculations
case of a maitensitic transformation for instance. As bainite is mainly composed of ferrite, its
contribution in the volume is measured via the ferrite lattice parameter 'aa-B,
In this equation, just as in the following one related to the intercritically annealed samples,
what is looked for is the value of ' / « - b ' (which is equal to 1 - I t
represents the
fraction of bainitic ferrite, while the interlamellar cementite will be included in the '
'
term. The part of cementite in a steel of such a composition is quite low.
The change in length 'Al' is the data measured by the dilatometer, while 7' is the length of the
sample : 10 mm. The different lattice parameters must be written as is :
Clfj^ — a^jQ
ay
-
(ayQ
Ci ferrile
+ C ] * Xl + C 2 * X J ) * OC austenite *
^
One should remember that x\ and X2 are the weight concentration of carbon and manganese in
the austenite, and c\ and ca are appropriate coefficients for the effect of the carbon and the
manganese. These elements have an influence on the size of the lattice parameter of the
austenite, in which the alloying elements have a higher solubility than in the ferrite lattice.
As the manganese stands in the iron lattice as a substitutional element, we may assume that it
diffuses slowly, and thus X2 is constant. On the contrary, the carbon is much more mobile, and
it obliges us to apply the following assumption : the cai'bon concentrates in the austenite. This
is why x\ must be written :
fr
!-ƒ«
On a mathematical point of view, it is interesting to note that a nice form of the solution of the
complete equation cannot be found easily. This is why the Newton-Raphson method is a good
choice for solving this equation.
Intercritically annealed samples :
A/ _ (2
lo
* al_, + 2 * f , _ , * a l , + f^_, * a%) - (2 * f„_, * a^,
+ f^_, *
)
3*(2*/a-;*4-7+/r-/*4-/)
'-ƒ ' indicates the "intercritical region". In this case it must be taken into account that there is
already some ferrite inside the sample before the isothermal transformation of the austenite
23
2. Material and metliods
Calculations
begins. This is why the term '2* f^_, *al_,\
which represents the intercritical ferrite, is
present in V' and Vo at the same time. Note that 'ƒ„_/' is a constant, either 0.1 or 0.55,
depending on the temperature ofthe intercritical annealing, 800°C or 750°C.
What is going to be calculated is ' fy_,',
which represents the fraction of (retained) austenite
and also the fraction of interlamellar cementite that is produced by the bainitic transformation.
Then it will be easy to find '
fa-B='^-fa-I
', the fraction of bainitic ferrite, by using this formula :
-fy-B
(10)
The details for the expressions of the lattice parameters are the same than in the case of the
completely austenitised samples.
2.3.2 Newton-Raphson
Newton-Raphson is the name of an iterated calculation technique that allows to find easily a
root of any equation, and it is especially interesting for equations that cannot be solved in an
analytic way. The equation that must be solved should be written in the following way :
F{x) = 0. The origin of the formula can be explained by the development of F(x) in a Taylor
series starting from a point xo.
(X-XQ)
F{x) = F{Xo) + {x-Xo)F'(Xo)+'
^^"^
2
F"(A-o)+...
(11 )
If this Taylor series is truncated after the term of the first order, and i f we are looking for a
root, i.e. F{x) = 0, we get
0=F(x)
= F(xo) +
( 12 )
(A--A-O)F'(XO)
and thus
^ - = ^ 0 - - ^
(13)
As the Taylor series was tmncated, the equation (13) is only an approximation of the solution.
A better approximation can be found by repeating the operation as suggested by the following
formula :
^M=Xi-T:7r^
(14)
24
2. Material and metixods
Calculations
This method is extremely powerful but it needs the evaluation of the value of the derivative
F{xi). This can be done easily in the following way :
^,.,) = Z < i ± f ) _ £ ( £ ^
,,,,
where £ is a fixed small number. Once the difference between
JCi+i
and x\ has come under a
defined threshold, the iteration is stopped. The user also has to define a starting value XQ ; this
one should be chosen close to the expected root in order to improve the convergence.
The Newton-Raphson method has been used in this work for the resolution of all the
equations involving the phase fractions, function of the lattice parameters, who are themselves
function of the phase fraction.
2.3.3 Matlab
Most of the calculations have been performed with the mathematical software MATLAB.
Indeed, it is very convenient for the handling of big data files such as those produced by the
dilatometer. A non-expert user can quickly manage with it as the programming language is
simplified in comparison with a language like Pascal. It is obviously very adapted to the
functional programming, i.e. one can define functions that will be used by different programs.
Another advantage is that no compilation is necessary so the programs can be modified pretty
easily. And a great interest of using Matlab is that one can present his results on very
functional and readable charts. Unlike with Microsoft Excell, for instance, the user does not
have to settle a lot of parameters in the picture, and, furthermore, the pictures do not need lots
of memory on the hard drive of the computer, because the file format is postscript.
25
3. Results
First measurements
3. Results
This chapter contains three types of results : dilatometric observations, calculations based on
the dilatometric data, and microstructural examinations. A l l of them are distributed
throughout four sections. Section 3.1 is about information on the low silicon steel that were
obtained thanks to gauging experiments. The second section ( Section 3.2 ) develops a
calculation based on the heating part of the gauging experiment. Section 3.3 , the biggest part,
contains examinations of the bainitic holdings as well as calculations. The chapter ends with
the section 3.4, which contains an analyse of a quite exotic dilatometric experiment.
3.1 First
measurements
This part comprises results that could be obtained by the analyse of the data of the first
gauging experiments. A slow heating to the austenitic region followed by a slow cooling
allowed to measure the characteristic temperatures, the expansion coefficients and the
transformation curves upon cooling and heating as well as at the equilibrium.
3.1.1 Characteristic temperatures
The first dilatometric experiment that was performed aimed to get basic information on the
material such as the temperatures A c l , Ac3, A r l and Ar3. For this, a sample was heated to
900°C at the rate of 3 K/min, which is the industrial practice, then it was kept at this level
during 30 minutes, and finally cooled at 3 K/min. The whole experience lasted 10h30 ; this is
sketched on Figure 10 hereunder.
120
Temoerature f-Cl
Figure 10: Dilatation ofa sample heated to 900°C at 3 K/min, then cooled at 3 K/min.
26
3. Results
First
measurements
One can properly see the phase transformation that takes place upon the heating as well as the
one that takes place upon the cooling. We can notice that, at POO-C, the sample is completely
austenitised. The second chart also represents the change in length, but this time as a function
of the temperature. Therefore it is possible to measure the characteristic temperatures.
"C
Acl
Ac3
Arl
Ar3
728
848
626
738
Table 5: Ciiaracteristic temperatures of tiie low-silicon steel.
'c' stands for 'citauffer' ('to heat' in French), and Y stands for 'refroidir' ('to cool' in French).
The temperature A2 is the one at which the iron becomes ferromagnetic when cooling down.
It is possible to measure it by looking at the curve that shows the electric power spent in the
coil as a function of the temperature (figure 11). As a matter of fact, there is a big step at
750°C that indicates the A2 temperature. The ferromagnetic transition demands a special
amount of energy, just like a fusion would do.
Power
50
40
30
20
10
0
0
200
400
600
Temoerature PCI
800
1000
Figure 11: Electric power spent in tiie induction coU upon iieating ofa sample.
As we ai-e dealing with the characteristic temperatures of a steel with a determined
composition, we are able to calculate the temperatures corresponding to the 'bainite start' and
the 'martensite start'. Here are Andrew's formulas ^^^^:
Bs = 830 - 270 * [C] - 90 * [Mn] - 37 * [ M ] - 70 * [Cr] - 83 * [Mo]
(16)
Ms = 539 - 423 * [C] - 30.4 * [Mn] -17.7 * [Ni] -12.1 * [Cr] - 7.5 * [Mo]
(17)
27
3. Results
First measurements
For our steel composition, we get
Bs =
652°C
Ms =
426''C
Those values agree with the TTT and CCT diagrams obtained from Bhadeshia's web site ^^''l
3.1.2 Expansion coefficients
Another important infomation that a dilatometer can bring is about the expansion
coefficients. It is very important to have accurate values for them since they enter into account
for all the calculations. Just by looking at the dilatation curve of this low silicon steel, one can
set his ideas about those coefficients. A deeper research will follow in the section 3.2.
Alpha and gamma
130
500
600
700
800
Temperature f^Cl
900
1000
Figure 12: Dilatation curve upon iieating ofa sample at 3 K/min.
Figure 12 shows the change in length as a function of the temperature during the heating. The
rings indicate the boundaries of the sections on which the expansion coefficient are
calculated. The derivative of this curve is plotted on the Figure 13. The mean values of the
expansion
coefficients are
calculated
between
the
boundaries,
and
we
get
:
aa = 17.14*10"^ K"' , for the ferrite, and Oy = 23.62*10"'^ K'" for the austenite. Of course, these
pai-ameters do not depend on the temperature here although they should. Oa is just a mean
value between 550°C and 725°C, and (Xyis between 850°C and 950°C. In the further
development, a more accurate coefficient will be obtained for the ferrite, but we will keep
23.5*10"'^
for the austenite.
28
3. Results
First
measurements
Expansion coefficients
30
20 Ferrite E C : 17.14 K^-1
Austdnite E C : 23.62 k^-1
^
10
o
O
Q.
X .
LU
-20
-30
500
600
700
800
Temperature f-Cl
900
1000
Figure 13: Measured expansion coefficient as a function of tlte teinperature. The given values
are for the averages on the regions defined hy the rings.
3.1.3 Transformation curves
This comprises the cooling part and the heating part. What is meant by 'cooling part' is the
transformation of the austenite into pearlitic ferrite during the experiment at 3 K/min. Figure
14 illustrates the phase transformation upon cooling. How can one measure the austenite
fraction that disappears during the cooling ? As explained earlier, we can do it with the lever
i-ule described in figure 9, or with a calculation based on the lattice parameters.
Cooling part
600
650
700
750
Temoerature 1=01
800
Figure 14: Dilatation curve measured upon cooling.
The result of the lever rule method is plotted on Figure 15. It is necessary to remember the
previously estabhshed characteristic temperatures: A r l = 626°C and Ar3 = 738°C. The tips of
the lever rule curve seem to be a little bit too much on the right.
29
3. Resuhs
First
measurements
Lever rule
600
650
700
750
Temoerature PCI
800
Figure 15: Calculated fraction of tiie austenite that disappears upon cooling. By tiie 'lever
rule' metiiod.
It is worth trying another Idnd of measurement for the phase fraction. The curve on Figure 16
is the result of a calculation that involves the lattice parameters : Theo Kop, from the
Materiaalkunde in TUDelft, proposed a program applied to the pearlitic transformation during
the cooling of a completely austenitised steel. The result seems in better agreement with the
measured values of A r l and Ar3. Both lever rule and Theo Kop's program show an angle in
the curve when the last 20 % of austenite fmally transform into pearlite.
Theo Kop's program
600
650
700
750
Temperature f=Cl
800
Figure 16: Calculated fraction ofthe austenite that disappears upon cooling. By a metiiod
based on the lattice parameters, and programmed by Theo Kop.
As a matter of fact, a quick calculation gives about 20 % of austenite when the eutectoïd level
^ ^ 0.16-0.02
I S reached ;
= 18.5%.
0.78-0.02
30
3. Results
First
measurements
After this quiclc loolc at the cooling part, the investigation continues with the heating part. The
ring in the inflexion on the curve of the Figure 17 indicates the end of the decomposition of
the cementite contained in the pearlite. The fraction of coming austenite can be calculated by
the lever rule method. It requires to draw linear extensions for both ferrite and austenite
(Figure 18), and then the equation (1) is applied. The result is displayed on Figure 19.
Heating part
700
750
800
850
900
Temoerature i-Cl
Figure 17: Dilatation curve upon iieating at 3 K/min.
750
800
850
900
Temoerature f^Cl
Figure 18: Application ofthe 'lever rule' metiiod upon heating. Extensions for the ferrite and
for tiie austenite have been drawn.
31
3. Results
First measurements
Lever rule
80
0
700
750
800
850
900
Temoerature l-Cl
Figure 19: Fraction of austenite calculated by 'lever rule' upon heating.
One should remember that A c l = 728°C and Ac3 = 848°C. These values are fitting quite well
with the chait that yields the fraction of austenite during the heating. One can still observe a
slight inflexion around 20 % of austenite. It corresponds to the end of the decomposition of
the pearlite.
The introduction of this thesis talks about an intercritical annealing, i.e. a thermal treatment in
which a steel is maintained for several minutes at a temperature comprised between A c l and
Ac3. The idea is to get, at the end of this intercritical annealing, a microstructure that contains
both ferrite and austenite.
It is worth remembering that most of the experiences performed in this research includes a
bainitic or a martensitic holding preceded by either a complete austenitising or an intercritical
annealing. It is therefore time to choose the parameters of these treatments.
The phase fraction depends on the chosen temperature, as can be seen on Figure 19. As this
curve corresponds to the heating, it is a good starting point for the choice of two intercritical
temperatures. I f the first one must give around 20 % of austenite, 750°C seems to be a good
choice. And i f we want more than 50 % of austenite, we can take 800°C as a second
intercritical temperature. Finally, the choice of an austenitising temperature must be done
according to the conventional rule : = Ac3 + 50 K. This is why 900°C was chosen.
When the distribution of the phases during cooling and heating at low rates is known, one
must check it with an MTData calculation. The relative positions of the curves with respect to
the equilibrium curve should fit.
32
3. Results
First
measurements
3.1.4 MTData calculation ofthe equilibrium
The MTData software allows to calculate the distribution of the phases at the equilibrium, just
on basis of the composition ofthe alloy ^^^1 The assumption intended by 'equilibrium' is that
every alloying element would have enough time and kinetic energy to diffuse from one phase
to another. For instance, all the carbon and the manganese would go in the austenite, while the
silicon would go in the ferrite.
Applied to our case, the crystallographic phases that must appear in the current calculation are
• ferrite : Body Centred Cubic.
•
cementite.
• austenite : Face Centred Cubic.
MTData; results
80
B C C A2
60
1 50
f8 40
FCC Al
.c
Q.
r
30
20
10
CEMENTITE
850
900
950
1000
1050
1100
1150
Temperature fK]
Figure 20: Output data of an MTData calculation of the equilibrium. Three phases are taken
into account: ferrite, austenite and cementite.
Figure 20 gives the concentration of unit cells of those phases as a fiinction of the
temperature. The number of cementite unit cells is very low, even under 950 K, because this
steel has a low carbon content (0.16 wt. %), and because the cementite unit cell is quite big
compared to the ferrite and the austenite cells. Beside this, it is obvious that the ferrite (at low
temperatures) has approximately two times more unit cells than the austenite (at high
temperatures). This result can be drawn in such a way that we get the percentage of austenite
as afiinctionofthe temperature. Indeed, this is the third curve of Figure 21.
33
3. Results
Calculation upon heating
80
I
60
ZJ
m
"5
§ 40
••8
^ 20
600
650
700
750
800
Temperature l°C]
850
900
Figure 21: Synthesis ofthe previously calculated curves for the austenite fraction.
The first curve in figure 21 is thefractionof austenite during the cooling calculated by Theo
Kop's program, the second one coiresponds to the cooling as well but calculated by the lever
rule, while the last curve corresponds to the heating, by lever rule. The approximate positions
of Acl, Ac3, Arl and Ar3 help to understand that the equilibrium curve is properly situated in
the middle with respect to the heating and the cooling curves.
A good thing would have been to calculate the para-equilibrium curve. In this case, the
assumptions are that the carbon can still diffiise perfectly, but the bigger atoms like
manganese and silicon, which are in a subtitutionnal solution position, cannot move, i.e. they
remain well distributed among each phase. The para-equilibrium is a situation that can be
reached in a few minutes, to the opposite of the complete equilibrium that requires several
hours, maybe several days. An intermediate assumption is closer to the reality of an
intercritical annealing of 10 minutes.
3.2 Calculation upon the heating data
This part deals with different calculations based on the dilatometric data coming from the
slow heating of the gauging experiment. The calculation method that is employed here is the
one developed in the section 2.3.1.2. First results are about the ferrite (3.2.1 and 3.2.2); while
others allow to produce a profile curve for the transformation of the ferrite into austenite
(3.2.3). An interest of these calculations is the fitting of several parameters, which are
summarised in the last part (3.2.4).
34
3. Results
Calculation upon heating
3.2.1 Calculation of the expansion coefficient of the ferrite
Here follows the results of a program that calculates the expansion coefficient of the ferrite.
By using the least square method, it fits a parabolic curve with the experimental heating curve
of the change in length as a function of the temperature. Figure 22 presents the experimental
curve as well as the fitting curve.
The second degree equation obtained for the parabola is :
-2.939 +0.131* r+0.00003747
=0
(18)
where the temperature is expressed in Celsius degrees. Let us remark that the second root of
this equation should give the temperature of the room where and when the experiment was
performed. As a matter of fact, the change in length is set to zero at room temperature. In this
case it was 22.4°C.
A correct result for the expansion coefficient can be found by dividing the 2 last terms by T.
a f.,. = (0.131 + 0.00003747 * T) * lO"'* [K"^]
=(13.1+ 0.003747* 7)* 10"^ [K"^]
Exp coeff of ferrite in R730
Temperature f°Cl
Figure 22: Fit ofa parabolic curve (red) to the heating part of a experimental curve (blue).
3.2.2 Lattice parameter of the ferrite and the cementite
In the previous paragraph, a value for the expansion coefficient of the ferrite was calculated.
Actually, our material is composed of ferrite with some cementite and the expansion
coefficient used in the calculation should take into account the presence of cementite. The
object of this part is the calculation of values for the initial (read at T = 20°C) lattice
parameters for the ferrite phase and the cementite phase. One way to do that is to plot an
35
Calculation upon heating
3. Results
experimental curve of a heating from 20°C to Acl, beside a second plot of a simulated change
in length using given coefiBcients.
The simulation can be done by using the formula (4) described earlier.
A/_F'(r)-Fo
As in this range of temperature the alloy is composed of ferrite and cementite, one needs to
know the proportion of the phases. For a steel containing 0.16 wt. C, if we consider that all
the carbon resides in the cementite, a quick calculation shows that there is 2.1 % of FesC.
VQ is the initial relative volume, i.e. it is equal to P when T = To = 20°C.
Here is an expression for
V'(T),
the current relative volume of the sample (it depends on the
current temperature) :
r{T) = 2* 0.979 *al{T) + y^* 0.021 * al{T)
(20)
where a a is the lattice parameter of the ferrite and ae is the lattice parameter ofthe cementite.
Simulation for ferrite
0
200
400
600
Temperature PC]
800
Figure 23: Fit ofa calculated dilatation airve (red) to an experimental dilatation curve
(blue).
Figure 23 holds the experimental dilatation curve (blue) and the calculated dilatation curve
(red and dotted). The deviation between them has been drawn on the Figure 24. Using the
parameters aao = 2.8830 A, aeo = 4.5234 A,
éeo = 5.0883 A,
ceo = 6.7426 A, we have a
correct estimation of the dilatation, since the difference remains small under A c l . This
calculation takes into account only two phases, so it is not correct anymore once cementite
starts to dissolve as austenite appears, i.e. once Acl has been reached.
36
3. Results
Calculation upon Pleating
Difference between experimental and simulation
^•4
c
-6
D)
C
^ -8
Ü
-10
200
400
600
Temperature f^Cl
800
Figure 24: Difference between tiie curves siiowed on the figure 23.
3.2.3 Transformation curve
Here comes an important result of this work. Using a calculation that implies the lattice
parameter and the change in length as a function of the temperature, it was possible to
determine the amount of austenite that appears during the heating of this low silicon steel.
3.2.3.1
Calculation
Only two phases can be taken into account in the calculation : ferrite and austenite. That is
why the temperature range on which it is acceptable starts at the end of the decomposition of
the pearlite, i.e. when the only phases that remain are ferrite and austenite.
Here is the expression for the change in length, adapted from equation (6) ^^'^:
M
2*f,_j*al+f,_j*a'-Vo
(21)
3*Va
YQ
/n
(•0
^
The -7 stands for intercritical. The idea is to calculate/y./ ( = 1 -fa-i) as the temperature and
the change in length are known. The lattice parameter of the ferrite is given by the following
equation :
= « « 0 * 0 + 13-04*10-^* r-l-0.003702*10-^*7^)
(22)
and the lattice parameter of the austenite is
(flyo+ci
*A-,
+C2*X2)*(1
+ 23.5*10-^*D
(23)
Equations (22) and (23) are identical to the equations (7) except for the values of the
expansion coefficients, which have been modified according to the above calculation. As the
37
3. Results
Calculation upon heating
solubility of alloying elements is much higher in the austenite, their concentrations have an
important effect on its overall lattice parameter. One way to take this effect into account is to
add a term including the weight concentration (x) multiplied by a coefficient (c). In the case of
this alloy, the important elements in solution are the carbon and the manganese. The weight
concentration of the carbon in the austenite is written xi while X2 stands for the manganese.
Usual values for ci and ca are 0.033 and 0.00095
^""^^1
Let us notice that the effect ofthe
carbon is much stronger than the effect of the manganese. The silicon prefers to go in the
lattice of the ferrite, so it does not exert a big influence on the lattice of austenite
An
important assumption that must be done was previously presented as the equation (8).
In the intercritical region, all the carbon concentrates in the austenite. Indeed, carbon is
gammagene, and diffusion is high enough at temperatures where austenite can appear, i.e.
above Acl, since the carbon atom is small and diffuses easily.
The solution ofthe complete equation can be found by using the Nev^on-Raphson method.
Let us look at the result for heating up to 900°C.
Annealing temperature : 900 "C
10
15
Time fminl
Figure 25: Representation of the end of a heating to 900°Cfollowed by an intercritical
annealing of 10 minutes
The chart of Figure 25 describes the heating to 900°C of a sample in the dilatometer. The
X-axis represents the time of the experience. As the heating rate is 100 K/min, 900°C is
reached after 9 minutes. The dotted blue line is the temperature. The continuous line is the
38
3. Results
Calculation upon heating
change in length of the sample. And the third line represents the fraction of austenite,
calculated with the equation (21).
Simulation for austenite
i
700
1
I
'
750
800
850
Temperature f-Cl
Figure 26: Calculated fraction of austenite that appears upon heating to 900°C.
This first result (showed again on Figure 26) was used in order to set parameters such as the
expansion coefficient of the austenite and the coefficients for the carbon and the manganese.
Let us look at the fraction of austenite as a function of the temperature. The upper part should
remain at 100 % and it should be flat. The first thing is to define an expansion coefficient for
which the slope of the upper part was equal to zero, and then define the coefficients c; and C 2
for carbon and manganese for which the plateau was set at 100 %. As a matter of fact, ci and
C2 are fitting parameters. The finally chosen values are c\ = 0.046, C2 = 0.00103 and oCy =
23.5*10"^ K-'.
Let us see now where goes this new "heating curve" with respect to the lever rule curves that
were talked about in the part 2.2.2.4. On Figure 27, the green curve is the one calculated
presently and that is plotted on Figure 26. Remember that the blue curves correspond to the
cooling part, the black one is the equilibrium and the red one is the fraction of austenite
during the heating calculated by the lever rule.
39
Calculation upon heating
3. Results
600
650
700
750
800
Temperature [°C1
850
900
Figure 27: Synthesis ofthe calculated curves for the austenite fraction as a function of the
temperature. (Same chart than this of the figure 21, but with the curve of the figure 26.)
Now that we have values for the fitting parameters, the calculation can be applied to a heating
up to 750°C followed by an intercritical anneahng of 10 minutes.
Annealing temperature: 750 °C
10
1200
12
14
Time fminl
Figure 28: Representation of the end ofa heating to 750"Cfollowed by an intercritical
annealing of 10 minutes.
On Figure 28, the dotted line still represents the temperature just like on Figure 25, the
continuous line is the change in length and the interrupted one is the fraction of austenite. One
can notice that once 750°C has been reached, there is still some transformation all along the
annealing of 10 minutes. At the end, the calculation of the equation (21) gives an austenite
amount of about 46 %. Remembering the assumption that all the carbon concentrates in the
40
3. Results
Calculation upon heating
austenite, the carbon concentration can be calculated with the equation (8). The result is
showed on Figure 29 and it stabilises at 0.35 wt. % C at the end of the annealing. This
number will have an important influence on the fiiture transformation(s) of this austenite,
since carbon slows down the formation of ferrite upon cooling
Annealing temperature : 750 "C
100
80
'I
0.8 £>
CT
O
Final fraction of
austenite : 46 %
60
w
0.6
(0
•B
c
40
0.4^
O
S
O
ZI
O
(O
Final [C] In
austenite: 0.35 %
20
O
0.2
LL
10
12
14
Time fminl
16
18
Figure 29: Representation ofthe calculatedfraction of austenite, as well as the carhon
concentration in the austenite during cm intercritical annealing at 750"C.
Annealing temperature : 800 °C
1201
'
,
,
'
I
i
1
I
5
10
15
20
Time fminl
Figure 30: Representation ofthe end ofa heating to 800"Cfollowed by an intercritical
annealing of 10 minutes.
The third heat treatment in the heating part is an intercritical annealing at 800°C. Results of
the calculation based on the equation (21) is showed on Figure 30. We can expect that there
will be more austenite formed by the end ofthe annealing than in the case of the annealing at
41
Calculation upon heating
3. Results
750°C. The curves of Figure 30 show obviously that more than 10 % of the fmal austenite has
been formed after the 800°C level was reached. According to the calculation, the fraction of
austemte should be 91 % at the end of the 10 minutes anneahng. That would mean an
austenite charged with 0.18 wt. % C, which is not very different from the overall
concentration (figure31).
Annealing temperature : 800 "C
100
Final fraction of
austenite: 91 %
80
0.8
o
c
1
60
0.6Bo
(0
c
ZJ
O
A0\
Final [C] In
austenite: 0.18 %
O
(0
0.4§
n>
O.2I
20
3
10
15
20
Time fminl
Figure 31: Representation ofthe calculatedfraction of austenite, as well as the carbon
concentration in the austenite on an intercritical annealing at 800°C.
In order to set the ideas concerning the effects of the different annealing treatments, one can
look at the pictures showed in the Figure 32 to 34. Remember that those resuhs are yielded by
the calculation based on the equation (21). Only the phase fractions presented hereafter will
be taken into account in the further calculations.
900 " C
Austenitised at 900°C during
10 minutes.
Carhon concentration in the
austenite: 0.16 %.
Figure 32
42
3. Results
Calculation upon heating
800 " C
Annealed at 800V during JO
minutes.
Carbon concenh'ation in the
austenite:
0.18%.
Annealed at 750V during JO
minutes.
Carbon concentration in the
austenite : 0.35%.
Figure 34
During the ulterior cooling, the ferrite will remain the same. What is going to interest us is
what will happen to the austenite in the continuation ofthe thermal treatment.
3.2.3.2 Microstructure analysis
The usual way to measure the fraction of phase in an intercritically annealed sample is to
quench it to the room temperature and practice image analysis on the metallographic
specimen. A sample was heated according to the temperature program described on Figure 35
and then quenched so that the austenite was completely transformed into martensite.
Time [mini
Figure 35: Heat treatment for the image analysis of a sample annealed at 750V.
43
Calculation upon heating
3. Results
After having followed the procedure described in the section 2.2.2, the final result for the
sample annealed at 750°C (picture 3) is a fraction of 44.2 % for the bright phase, which is the
martensite. The standard deviation for the whole data is 2.77 %. The fraction of austenite is
therefore around 44 % as well.
1.39kX
UCillM
20kU HDilShird
L.
S; r r 7 P.00081
_
Picture 3 (RR7): SEM micrograph ofa sample intercritically annealed at 750°Cfor
10 minutes, then quenched to room temperature. Magnification : 1400 X.
4.09kX
10un
20kU UDUemm
8:06000 P i d S a s a
Picture 4 (RR8): SEM micrograph ofa sample intercritically annealed at SOO^Cfor
10 minutes, then quenched to room temperature.
44
Calculation upon heating
3. Results
The SEM micrographs of the sample annealed at 800°C were also analysed with Visilog, The
result obtained yields 58,1 ± 5.2 % of martensite, i.e. austenite before the quench,
3.2.4 Fitting parameters
Here is a table that summarises the whole set of coefficients that have just been defined. It
includes the expansion coefficients, the lattice parameters, and the coefficients c; and C2 of the
equation (23), These numbers are mostly inspired from a paper of Dyson and Holmes
and
have been refined to fit the calculation. In a first time, the assumption is that the composition
of the alloy is the one given by Hoogovens, i.e. from the table 1,
Mn
Si
C
1,5
0.4
0.16
Wt. %
Table 1: Composition given by Hoogovens.
For this composition, the parameters are presented in the Table 6 :
= 2.883 [A]
atO°C
a
= (13.1 +0.003747*T)*10"^
T expressed in °C
ClQO
= 4.5234 [A]
at 20 °C
bm
= 5.0883 [A]
at 20 °C
Ceo
= 6.7426 [A]
at 20 °C
a
=
Ferrite :
Cementite :
5.311*10'^
-
1.942* 10'^*(T+273)
+ T expressed in °C
9.655* 10"'^*(T+273)^
Austenite :
fl-yO
= 3.5972 [A]
a
= 23.5*10""
Cl
= 0.046
C2
= 0.00103
at 25 °C
Table 6: Ciwsen parameters when the composition is defined by the tahle 1.
The parameters have also been calculated assuming the composition that has been measured
in Delft, i.e. from the table 2.
Wt. %
Mn
Si
C
1.47
0.28
0,14
Table 2: Composition measured at the TUDelft.
45
3. Results
Bainitic
transformation
In this case, the only parameters that are different from those of the Table 6 are cj and C2>
from the equation (23). As a matter of fact, these are the only parameters who depend on the
concentration of the alloying elements. In this case, they must be set at:
Cl
0.052
C2
0.0011
Table 7: Parameters for the influence oftite carbon and the manganese according to the
composition defined in table 2.
Nevertheless, it seems more appropriate to work with the composition of the table 1 because
of the complexion problems that may have appeared during the chemical analysis with the
ICP.
3.3 Bainitic
transformation
Firstly, the dilatometric curves will be observed and compared according to the thermal
treatment applied to the samples. In the second part, there will be a short analysis of the
microstructures obtained for the different annealings. And finally, we will see the results
obtained by the calculations presented in the section 2.3.1 and applied to the data measured in
3.3.1.
3.3.1 Isothermal holdings
750
120|
^,00-
QlZ
0
'
to 300
'
<
^
,
.
,
1
10
20
Time fminl
30
40
Figure 36: Typical dilatation curve of an experiment that comprises a bainitic holding.
46
Bain itie transforma tion
3. Results
As explained in the introduction of this text, the usual way to stabilise austenite down tb room
temperature is to impose a bainitic transformation. The mechanism is favourable to the
survival of this phase out of its equilibrium. This is why I have been inquiring on the bainitic
transformation for the composition previously described.
In a first approach we will consider that nothing happens during the quench and that the
transformation occurs only during the 15 minutes holding. The figure 36 shows which part of
the measurement is interesting for us. The part where the curve is thicker corresponds to the
isothermal holding, in this case, at 300°C. What defines the beginning of the holding is the
moment when the temperature has been stabilised at the very end of the quench. This point is
indicated on the figure 37.
5501
'
Quench to 300-C
'
'
250
200'
1054
'
1055
—'
1056
1057
Time fsecl
'
1058
'
1059
Figure 37: Sitape ofthe end ofa quench. The ring defines the moment when the isotlienncd
transformation starts to be measured.
For each of the three different annealing temperature, the curves corresponding to every
isothei-mal holding have been reported on one chart with a logarithmic time scale on the Xaxis. The figure 38 contains the curves of the change in length during 15 minutes
corresponding to the completely austenitised samples, i.e. austenitised at 900°C. The absolute
vertical position of a single curve should not be taken into account as there may have been
moves due to vibrations during the experiment. What is important is the absolute increase
between the beginning and the end of the isothermal holding, and this is illustrated on the
figure 39.
47
3. Results
Bainitic transformation
Series 900 "C
7
o
*
c
0)
C3)
c
(0
x:
O
10"
10
10
10^
Time fsecl
Figure 38: Set of dilatation curyes corresponding to the isothermal transformations that takes
place at different holding temperatures in the case of completely austenitised samples.
Series 900 "C, fielding: 15 mln
50
40
f
30
f
20
c
(0
Ü—fl-10
100
200
300
400
Holdina temperature f C l
500
Figure 39: Total variation ofthe length upon the 15 minutes holdings showed on figure 38.
The results showed on the figures 38 and 39 are not like one could expect. Indeed, the
absolute change in length should increase with lower holding temperatures, because the
thermodynamical driving force for the transformation is increased as well. This behaviour is
observed here only for the holding temperatures of 450°C, 400°C and 350°C.
On the other hand, one can notice that the transformations appearing for the holding
temperatures of 300°C and under are very light in comparison to the upper levels. The fact is
that, in those cases, a martensitic transformation has occurred during the quench, and the light
increase of the change in length during the 15 minutes is due to a precipitation of carbides in
48
3. Results
Bainitic transformation
the martensite ^ ' . Generally, the incoherence is explained by the existence of phase
transformations during the quench. This problem will be discussed in the fourth chapter.
Here comes the measurements on the isothermal transformations of the intercritically
annealed at 800°C (figures 40 and 41).
Series 800 "C
Time fsecl
Figure 40: Set of dilatation curves corresponding to the isothermal transformations that takes
place at different holding temperatures for samples intercritically annealed at 800V.
Series 800 "C, holding: 15 min
15r
<
o
10
•6)
c
ffl
0)
O)
c
ro
x:
O
100
200
300
400
Holdina temperature f C l
500
Figure 41: Total variation of the length upon the 15 minutes holdings showed on flgure 40.
On a first look, the result seems to be quite equivalent to the case of the samples completely
austenitised (Figure 38 and Figure 39). But actually,ft)rthe highest holding temperatures, the
relative variation is about 2.5 times less (look at the scale ofthe charts) : for the 15 minutes
isothermal transformation at 350°C, the variation is here of 13*10"^ and it was of 30*10"^
49
3. Results
Bainitic transformation
previously. This is surprising when one thinks that the starting amounts of austenite were of
+91 % (according to the calculation) and 100 %, i.e. not very different. The explanation for
such a difference in relative variation of the change in length must be given in the fourth
chapter of this paper (section 4.2.1).
What about the other intercritical annealing : 750°C ?
Series 750 °C
80
^
60
o
*
t
c
«J
c
40
0)
^
20
m
JZ
Ü
10"'
io''
10°
io'
Time fsecl
Figure 42: Set of dilatation curves corresponding to the isothermal tiansformations that takes
place at different holding temperatures for samples intercritically cmnealed at JSOV.
Series 750 °C, fiolding : 15 min
25 I
'
'
i
.
r-
<
o
•6)
c
10¬
0)
O)
c
5
.
5Qj
l l l l
100
M i l l
I l l l l
I l l l l
I l . l l
200
300
400
500
Holdina temoerature f C l
I
I
600
Figure 43: Total variation of the length upon the 15 minutes holdings showed on figure 42.
One can still observe the gap between the high and the low holding temperatures, but this
time, 300°C belongs to the group that allows an important transformation. It means that the
50
Bainitic
3. Results
transformation
temperature for the martensitic transformation (Ms) has decreased, hideed, i f we repeat the
calculation of Ms with Andrew's equation (17), but this time for an austenite that contains
0.35 wt. % C, we get 290°C. This value is more than 100 K under the Ms of the completely
austenitised steel. Another interesting remark is that the mean variation of the change in
length all along the 15 minutes is slightly higher in this case than for the annealing at 800°C
although there was less austenite to be transformed (45 % instead of 90 % ) .
The charts of the figures 44 to 46 indicate the variation of the change in length for the first
minute of the isothermal holding, When compared to the chaits for 15 minutes, it becomes
obvious that most of the transformation happens during that first minute, as there is almost no
difference between the results presented in these charts and those previously showed.
Series 900 -C, holding: 1 mln
Series 800 -C, holding: 1 mln
50
40
30
20
10
o
—&•
-10
100
200
300
400
Holdina temoerature f-Cl
500
100
Figure 44
200
300
400
Holdina temoerature f»Cl
500
Figure 45
Series 750 °C, holding : 1 min
Ciiange in lengtii after 1 minute of isotiiermal
iioldingfor :
Austenitised samples (figure 44).
Samples annealed at 800°C (figure 45).
100
200
300
400
500
Holdina temoerature f-Cl
Samples annealed at 750°C (figure 46).
600
Figure 46
What is worth to be noticed is the value of the "750°C" series corresponding to the holding
temperature of 300°C ( Figure 46). It is quite lower for a holding of 1 minute (7.5*lO"'^) than
51
3. Results
Bainitic transformation
for the 15 minutes (18*10"'*). It means that that treatment causes an isothermal transformation
more widely spread in the time. Indeed, 300°C is still a little bit higher than Ms, and it is cold
enough to slow down a diffusive transformation.
3.3.2 Microstructure analysis
What is the effect of the annealing temperature on the bainitic transformation ? It is visible by
the shape of the dilatation curve during the isothermal holding. A comparison between three
experiments might help. Three bainitic holdings at 350°C have been plotted on the Figure 47:
Quenches to 350°C
-20'
0
'
10
•
20
Time fminl
•
30
'
40
Figure 47: Dilatation curves for experiments with different intercritical annealing
temperatures (750"C, 800T and 900V), followed by an isothermal holding at 350V.
The three experiments showed on the figure 47 and, separately, on the figures 48 to 50 differ
by their austenitisation conditions and have the same temperature of isothermal holding:
350°C. The annealing temperatures were 900°C, 800°C and 750°C. What is interesting for us
is the shape of the curve after the quench; the three charts of the figures 48 to 50 have the
same scale, so that the curves can be properly compared.
Independently of the amplitude ofthe variation (that has been partly explained earlier), let us
look at the slope of the curves, which gives an idea of the transformation rate. It is very fast in
the case of the completely austenitised sample, whereas the higher carbon content of the third
sample tends to slow down the isothermal transformation.
Firstly, the rate of the transformation of austenite into ferrite is very dependent on the carbon
concentration. Furthermore, it is not necessarily the same phase that forms since Ms depends
on the carbon concentration in the austenite. After a full austenitisation at 900°C, Ms is
around 400°C, and this means that the phase formed at 350°C is martensite. On the contrary,
after an intercritical annealing at 750''C, Ms is around 300°C.
52
3. Results
Bainitic
transformation
Microstractures corresponding to tlie three different treatments are visible on the pictures 5 to
7. One can see on the picture 5 that the austenitising treatment has increased the grain size
from 5 pm to about 10 pm (It can be seen even better on the picture 11). On the other hand,
the microstructures showed on the pictures 6 and 7 look quite hke those of the pictures 4 and
3 respectively, in what concerns the phase proportions. But the microstructure visible on the
picture 7 is decomposed bainite, since the bainitic holding was performed at 350°C and Ms is
around 300°C.
From SOO^C
20
22
Time fminl
24
22
Time fminl
Figure 48
24
Figure 49
From 750^0
Dilatation curves of experiments witii an
isotiiermal Jiolding at 350"C, following a
quench, from:
900°C (figure 48)
SOO^C (figure 49)
20
22
Time fminl
24
750''C (figure 50)
26
Figure 50
53
3. Results
Bainitic
transformation
Picture 5 (R935): SEM micrograpii ofa sample austenitised at 900''Cfor 10 min, then
quenched to 35 "C and held at that temperature for 15 min.
Picture 6 (R835): SEM micrograph ofa sample austenitised at 800°Cfor 10 min, then
quenched at 350''C and held at that temperature for 15 min.
54
Bainitic
3. Results
transformation
Picture 7 (R735): SEM micrograpii ofa sample austenitised at 750°Cfor 10 min, then
quenched at 350°C and held at that temperature for 15 min.
3.3.3 Calculation on the bainitic holding
The section 2.2.1 has already described the equations that will be used in this part.
Equation (6) will be applied to the completely austenitised samples, while the equation (9)
stands for the intercritically annealed samples. The calculation is now possible because the
dilatometric data is available since the section 3.3.1, and also because parameters have been
summarised in the section 3.2.4.
gOOi'G to 450^C
900^0 to 450^0
Figure 51: (a) Calculated fraction of austenite that disappears during the isothermal holding
at 450°C, following a quench from 900''C. (b) Calculated fraction of bainitic ferrite that
appears during the isothermal holding.
55
Bainitic
3. Results
transformation
Figure 51 shows the result for the isothermal transformation at 450°C, coming from 900°C.
One should be aware that the scale on the Y-axis of the two charts are different.
Unfortunately, this result is obviously wrong. It is simply impossible to keep more than 60 %
of untransformed austenite at 450°C with this kind of steel. The reality is that there are not
anymore 100 % of austenite at the beginning of the isothermal holding. We will try to solve
that problem in the fourth chapter.
What about the result for a sample intercritically annealed at 800°C ? (Figure 52)
SOO^C to 4 0 0 S C
800^0 to 400^0
Time fsecl
Time fsecl
Figure 52: (a) Calculated fraction of austenite titat disappears during tiie isotiiermal holding
at 400°C, following a quench from 800''C. (b) Calculated fraction of bainitic ferrite tiiat
appears during the isothermal holding.
Although 400°C is very far from the austenitic region, the fraction goes from 90 % to hardly
68 %. The problem is the same as in the case of the austenitised samples : there was not 90 %
of austenite at the end of the quench. It seems that it is worse in this case : the calculation
foresees the formation of 23 % of bainitic ferrite versus 37 % for the previous experience.
At last, there is the isothermal transformation of a sample annealed at 750°C (Figure 53). The
left picture suggests that there is a little bit more than 10 % of austenite remaining. But do not
forget that it includes the cementite that is produced by the bainitic transformation. It is very
acceptable in comparison to the two previous results; it seems that these data do not meet the
same problem, i.e. some transformation of the austenite during the quench.
Therefore, the experiments of the series "750°C" are chosen in order to draw a TTT diagram.
56
3. Results
Intercritical annealing at 750 "C : test
ySO^^C to 350^0
10
Time fsecl
750^0 to 350=0
10
10
Time fsecl
10'
Figure 53: (a) Calculated fraction of austenite tiiat disappears during tiie isotitennal holding
at 350°C, following a quench from 750°C. (b) Calculated fraction of bainitic ferrite that
appears during tiie isothermal holding.
3.4 Intercritical annealing at 750-C: test
This section is based on the study of a very particular experiment: the idea is to carry out an
intercritical annealing at 750°C, but this time, coming from high temperatures. In a first time,
the experiment is described, then the dilatometric results are presented, and the microstructure
photographs come at the end.
3.4.1 Introduction
Sybrand van der Zwaag has suggested a strange experiment: why not compare an intercritical
anneahng that would follow a heating from room temperature as usual (figure 54), with an
intercritical annealing that would be preceded by a full austenitisation at 950 °C (figure 55) ?
R735
20
30
Time [mini
40
50
Figure 54: Classical heat treatment for the production of TRIP-aided multiphase steel. The
intercritical annealing at 750°C is peiformed coming from low temperatures.
57
3. Results
Intercritical annealing at 750 °C: test
There are several ideas hidden behind this experiment:
• To compare the amounts of austenite contained in the samples at the end of the intercritical
annealing.
• To compai'e the growth of the austenite grains inside a ferritic matrix with the growth of
the ferrite grains inside an austenitic matrix.
• To observe the influence of the starting microstructure on the kinetics of the bainitic
transformation.
• To look at the grain size of the austenitised samples.
R9735
1000
_
l
800
a>
a
g.
600
§
400
200
0
0
10
20
30
Time [mln]
40
50
Figure 55: Heat treatment witli an inteixritical annealing at 750"C peifonned coming from a
iiiglier temperature.
Another way to understand this experiment is to look at the figures 56 and 57. During heating,
when the dilatation curve crosses the dotted line at 750°C, the transformation has already
begun. Whereas when cooling, the formation of ferrite hardly begins at 750°C. Now, what
happens during the 10 minutes of annealing ? One may think that the volume fraction of
austenite is going to evolve towards the equilibrium described by the MTData curve in the
section 3.1.4.
58
3. Resuhs
Intercritical anneahng at 750 V: test
Heating and cooling at 3 K/min
600
650
700
750
800
Temperature \°C]
850
900
Figure 56: Dilatation heating curve (red) and cooling curve (blue). The vertical dark dotted
line shows the temperature of the intercritical annealing.
Ar3
100
'
^ r
Ac3
y
I/ /
80
i
60
^—
O
I
Cool no/
Heatinc
40
(a
20
AH
600
650
700
750
800
Temperature ["Cl
850
900
Figure 57: Distiibution ofthe phases during cooling (blue) and heating (red). The dark
dotted curve shows the temperature ofthe intercritical amiealing.
3.4.2 Dilatometry
Before we observe the bainitic transformations, let us look at the annealings. The charts ofthe
figure 58 and figure 59 show that although the annealings are performed at the same
temperature (750°C) and for 10 minutes, the difference between the dilatations ofthe samples
is quite important. Afirstexplanation for this deviation is that the phase distributions at 750°C
is very diflFerent in the two cases. A quick look at figure 57 teaches that the sample that is
cooled to 750°C contains a lot more austenite than the one that is heated to 750°C. Indeed the
59
3. Results
Intercritical annealing at 750 "C: test
10 minutes holding cannot re-establish the equilibrium, And since the austenite lattice is more
compact than the ferrite's, the sample with a lot of y phase presents a lower dilatation.
1401
'
<
'
'
^
^
,
Time fminl
Figure 58: Dilatation curves of the experiments described by the figures 54 (red) and 55
(blue), but it is stopped at the end of the intercritical annealings.
1000
Figure 59: Dilatation curves ofthe experiments described by the figures 54 (red curve) and
55 (blue curve), but it is stopped at the end of the intercritical annealings.
Let us look now at the bainitic transformation that follows these intercritical annealings. The
figure 60 contams the complete curves of the experiments presented on the figure 54 and
figure 55. The transformations that occur at 350°C after the quench have really different
profiles. While the usual treatment, i.e. without austenitisation, yields a quite slow
transformation, the other one produces a very fast reaction. The reason for this resides in the
mutual austenite concentrations. Indeed, since the y phase ofthe austenitized sample is larger.
60
3. Results
Intercritical annealing at 750 °C: test
it has a lower carbon concentration than the y phase of the other sample. And it is well known
that the the higher the carbon content in the austenite, the more the phase transformations are
slowed down
3.4.3 Microstructure
120
^100
<
o
^ 80
s:
1
c
60
O)
ro
.c
Ü
40
20
0^
0
10
20
30
Time [mini
40
50
Figure 60: Complete dilatation curves ofthe experiments described by figures 54 and 55.
Lookmg at the micrographs will help to compare the microstructures : in the first case, the
austenite has grown in the ferrite, and in the second case, the ferrite has grown in the
austenite. The microstructure of a sample intercritically annealed at 750°C has already been
showed on the picture 3 in the section 3.2.3.2 devoted to the calculation during the heating.
The austenite has nucleated on the grain boundaries of the pearlitic microstructure. The
measurements showed that the sample contained around 44 % of austenite at the end of the
annealing. It was investigated by both calculation and image analysis.
Picture 8 that follows represents the sample that has undergone the treatment described by the
figure 55. This picture is magnified only 65 times so that one can see the huge spots
measuring nearly 1 millimetre. The microstmchire between the spots is banded, just Hke the
one ofthe original samples received from Hoogovens.
The big spots, visible on picture 8, measure more than 500 pm. They are fiilly martensitic
with very large grains measuring sometimes more than 50 pm long (picture 10). On the other
hand, the banded structure, which can be seen besides the martensitic spots on the picture 8,
contains a mix of ferrite and martensite in similar proportions. Its grain size remains around
10 pm, according to the picture 9. This is more or less the same gram size than that ofthe
samples completely austenitised and quenched under Ms (picture 5). It is obvious that
everything that is martensite on the pictures was austenite before the quench to 350°C.
61
3. Results
Intercritical annealing at 750 "C : test
Picture 8 (R9735): SEM micrograpii at low magnificence of tiie sample tiiat lias undergone
tiie treatment defined on tiie Figure 55.
Picture 9 (R9735): SEM micrograph at high magnificence ofthe banded structure visible on
the picture 8.
62
Intercritical annealing at 750 °C : test
3. Results
• •: S i l l
/ • ' I / - , ;•
^ •'!••
i f , -Kl
Picture 10 (R9735): SEM micrograph, at high magnificence ofa martensitic spot seen on the
picture 8.
63
4. Discussion
Heating
calculation
4. Discussion
The first section of this chapter (4.1) contains an interpretation of the calculation applied on
the heating curve, which was developed in the section 3.2. It is foUowed by a short
explanation of the results of the isothermal holding, and an investigation on the encountered
experimental problems (4.2). The third point presents one of the most important outcome of
this work : a TTT diagram (4.3), while the fourth part tries to retire information from the
failed experiments (4.4). Another calculation method is presented in 4.5. The results of the
exotic experiments on the intercritical annealing are discussed in 4.6. Finally, the appropriate
ways for stabilising the austenite are presented in 4.7.
4.1 Heating
calculation
The results obtained in section 3.2 are summarised in Table 8.
Calculation
Image analysis
Annealing at 750°C
46%
44.2 ± 2.8 %
Annealing at 800°C
91 %
58.1 ± 5 . 2 %
Table 8: Fraction of austenite calculated or measured at tiie end of tiie intercritical
annealings.
For the sample annealed at 750°C, the results of the calculation and the image analysis seem
to agree quite well. Since the image analysis technique is accurate with a standai'd deviation
of 2.77 % on 50 pictures, this is a good result.
For the sample annealed at 800°C, the resuh ofthe image analysis yields 58.1 ± 5.2 % against
91 % for the calculation. A possible explanation for that great difference is that a part of the
austenite has transformed into ferrite before Ms was reached. Looking at the curve that
describes the quench (Figure 61), one can observe a slight hump in the slope, which could
correspond to a partial transformation of austenite into ferrite
This theory is confirmed by microstructural observations of the picture 4. Indeed, there are
hints of ferritic laths peipendicular to the grain boundaries. This is the typical aspect of the
Widmanstatten ferrite, which is nothing else than ferrite formed during a fast cooling. This
64
4. Discussion
Bainitic
transformation
would explain the fact that the measured fraction of austenite is smaller than the calculated
fraction since a part of it would have transform into Widmanstatten ferrite during the quench,
1401
'
Quench from 800
'
.
^
1
120¬
.
|iooi
£
80-
§
60 •
ö
40
ro
I
20-
O
0•
.201
1075
'
1080
'
'
1085
1090
Time fsecl
'
1095
i
1100
Figure 61: Dilatation curve sixowing tlxe pitase transfonnation titat lias occurred during a
quench from 800 "C to room temperature.
A question that might be asked now is : "Why would there be a transformation during the
quench in the case of the annealing at 800°C and not for 750°C ?", The explanation is that, in
the first case, the carbon concentration in the austenite is high enough to prevent a
transformation before Ms has been reached. In the second case, the cooling rate is not high
enough to get rid of the formation of ferrite. According to the equation (8), the austenite of the
samples annealed at 750°C contain about 0.35 wt. % C, against 0.18 wt. % C for the samples
annealed at 800°C.
4.2 Bainitic
transformation
This section presents an interpretation of the results that were obtained in the section 3.3.
Once the main problem is localised, it will be deeper investigated, with eventually the help of
a CCT diagram. The last point of this section deals with the inaccuracies encountered with the
stability of the calculation presented in the section 2.3.1.2.
4.2.1 Synthesis of the results on the isothermal holdings
In the section 3.3.1, a big difference was measured between the change in length of the series
900°C (figure 39) and the series 800''C (figure 41) during isothermal holding, although the
amounts of austenite that had to be transformed were close (100 % and 90 %). The average of
the relative change in length was measured to be about 2.5 times less in the second case. This
65
4. Discussion
Bainitic
transformation
problem can be explained by the fact that the quenches were not perfect, i.e. there were
transformations during the quenches. In the case of the annealing at 800 °C, there was more
transformation during the quench than in the case of the complete austenitising (900°C).
Indeed, i f there is a big fraction transformed during the quench, the variation measured on the
isothermal holding, which corresponds to the continuation of the transformation, is small.
Another question evoked in the section 3.3.1 was : why is the change in length measured for
the 750°C series (during isothennal holding (figure 43)) bigger than the change in length
measured for the 800°C series (figure 41) although in this second case there is twice as much
austenite to be transfomed ? The answer is that a big part of the austenite was transformed
during the quench in the 800°C series, and nothing was transformed upon quenching in the
750°C series. The present section aims thus to investigate on what happens during the quench.
The strange measurements of the section 3.3.1 have induced strange results in the calculations
of the section 3.3.3. The results of the calculation applied to the dilatometric data of the
isothermal holding are presented in the charts of the figures 51 to 53. It is deceiving for the
series 900°C and 800°C, but it is acceptable for the series 750°C. Indeed, the proportions of
stabilised austenite for the two first series are exaggerated. The problem resides in the fact
that the program assumes that there is respectively 100 % and 91 % of austenite at the
beginning of the isothermal holding. There was actually much less austenite left in those
samples at that moment. Only the last series (annealing at 750°C) seems to present more
acceptable results for the calculation. The following part attempts to give an explanation for
this phenomenon.
4.2.2 Quenches
Most of the experiments that were perfonned on the dilatometer BAHR805 presented a
quench in their temperature program. One important advantage of that machine is that the
sampling rate can be chosen up to 1000 Hz. When applying such a high sampling rate to the
quench, it finally allows to draw a very accurate curve that describes the quench.
In the set of the experiments performed, there are three different types of quenches, depending
on the starting temperature : 900°C, 800°C or 750^0.
66
Bainitic
4. Discussion
R910
transfonnation
From 900=C to100=C
120
120 r
1140
1150
1160 1170 1180
Time fsec]
1190 1200
200
400
600
Temperature [-C]
800
1000
Figure 62: Dilatation curve ofa quench, from 900 °C showed (a) with the time as X-axis,
(b) with temperature as X-axis.
The charts of figure 62 represent the same quench from 900''C to 100°C : the first one with the
time on the X axis and the second one with the temperature. An important transformation
occurred ai-ound 600°C during the cooling. This is of course a problem for someone who
wants to use that experiment to draw a TTT diagram. Actually, the cooling rate was not high
enough. This is to be compared with quenches performed from intercritical temperatures.
From 800^0 to 200^0
R820
1078
1080 1082 1084 1086 1088 1090
Time [seel
200
400
600
Temperature f-C]
800
1000
Figure 63: Dilatation curve ofa quench from 800°C showed (a) with time in abscise axis, (b)
with temperature in abscise axis.
Figure 63 show a quench performed from 800°C to 200''C, where there should be about 90 %
of austenite at the beginning. An important transformation is visible around 700''C, which is
sooner than in the case of the completely austenitised sample (figure 62). We will see later
that those transformations at 600°C and 700°C are not exceptional phenomenons. Another
67
Bainitic
4. Discussion
transformation
characteristic of that quench is the existence of a plateau at the very beginning of the quench,
and it is visible on the second chart.
Finally, an analyse of the quench described by the figure 64, going from 750°C to 100°C,
shows that it does not present any transformation before Ms (290°C) is reached. (Therefore
that series could be used for the drawing of a TTT diagram.)
R710
From 750»C to 100=0
Time [sec]
Temperature [°C]
Figure 64: Dilatation cwve ofa quench, from 750''C showed (a) witli time in abscise axis, (b)
with temperature in abscise axis.
One should remark that, alike the previous experiment, there is a plateau at the top of the
slope on the second chait. Those plateaux are due to the effect of a temperature gradient from
the core of the sample to its surface. When the helium blow begins, the surface is cooled
earlier than the whole sample. And, as the thermocouple is welded on the surface, it indicates
a lower temperature than the real overall temperature, which is naturally related to the exact
change in length. Anyway, this artefact has no awkward consequences for the use of the data.
AU the considerations established here are based on the analyse of several experiments for
each one of the three cases.
4.2.3 C C T Diagram
It has been stated that, for two series, the main problem in the quenching was the too low
cooling rate. CCT diagrams are the appropriate tools to measure the critical cooling rate of a
specific steel composition. The following diagram has been released by a program written by
Pieter van der Volk, from the Materiaalkunde, TUDelft. The program is not finished yet, so
the output (Figure 65) may look a little bit rough, nevertheless, the aim is achieved.
68
4. Discussion
Bainitic transformation
C C T diagram
0
1
2
3
loQ(Timefsecl)
4
5
Figure 65: CCT diagram for the composition described in table 1 and assuming that the
material has been austenitised at 900"C.
As usual, CCT diagrams are made for completely austenitised steels; in this case : at 900°C.
Unfortunately, it does not produce CCT diagrams for steels that were intercritically annealed.
The diagram is composed of four Hnes: FS (ferrite start), B S (bainite start), P E (pearHte end),
(BE : bainite end). The user has to fmd his way through it by redrawing the curves with
thicker plots. The program also calculates the position of a nose before which the ferrite
cannot form. For this composition, the nose is at 6 0 I X and after 2.9 seconds.
The critical cooHng rate values then: CCR -
— =
2. .951
101.4 K/sec. There is no information
about the critical cooHng rate required to get rid ofthe formation of bainite.
4.2.4 Stability of the calculations
The results presented in the section 3.3.3. were yielded by a matlab program. It is important to
know how the result can be influenced by a slight change in the input data. The figures 66
to 68 show the calculated fraction of bainite for three different input parameters. The three
charts contain the so-called fractions of bainitic ferrite that appear during the isothermal
transformations at 350°C of samples quenched from 750°C, 800°C and 900°C respectively.
The curve in the middle is the one corresponding to the parameters of the table 6, while the
upper curve corresponds to an increase of 0.1 % of the austenite lattice parameter, and the
lower curve corresponds to an increase of 0.1 % of the ferrite lattice parameter.
69
4. Discussion
TTT diagram for the annealing at 750"C
Effect of a 0,1% variation on the 900 series
Effect of a 0.1 % variation on the 800 series
0.25
10
10
10"
Time fsecl
10"
Time fsecl
Figure 66
Figure 67
Effect of a 0.1% variation on the 750 series
Calculated fi-action of bainiticferrite formed
during an isothermal holding following an
annealing at 900°C (flgure 66), 800V (flgure
67), 750V (flgure 68). The dark curve is the
original, while the blue one corresponds to an
increase of 0.1% of the austenite lattice
parameter, and the red curve corresponds to
an increase of 0.1% ofthe ferrite lattice
10
parameter.
^.
,
, 10'
Time fsecl
Figure 68
As one could expect, an artificial increase of the lattice austenite parameter induces an underestimation of the amount of this phase in comparison to the ferrite, which is therefore over
estimated. And, logically, an increase of the ferrite lattice parameter has the opposite effect.
What is very important to note here is that a slight change (0.1 %) in the input data can create,
for a holding of 15 minutes, a variation of more than 5 % in the result. As a matter of fact, this
calculation technique based on the lattice parameters is unstable, i.e. the error is quickly
amplified. It means that it is dangerous to rely on the results without checking it by a different
characterisation technique. This is a strong limitation to this type of calculation, and it is
essential to be aware of it.
4.3 TTT diagram for the annealing at 750°C
It was demonstrated in section 4.2 that the data corresponding to an intercritical annealing at
750°C were relevant to draw a TTT diagram since no transfonnation occurred during the
quench.
70
4. Discussion
TTT diagram for the annealing at 750°C
Isothermal transformations
-
1 -..-»rtÉ550°C
--'-s^^^W500°C
-
Jf
/
n
/ /
•
U
// /
-
4500C
/_».<*^4oo°c
/
/
300°C
y250''C
//:iC*-><^,»200°c
^ ^ . — • - ^ ' ^ ^ l 50°C
VVWJyl»...
00°C
'
10°
10'
10
Time [sec]
Figure 69: Calculatedfractions
of transformed phase during isothermal holdings at 10
different temperatures The material was intercritically annealed at 750V.
Figure 69 siiows tlie curves of tliefractiontransformed for the different holding temperatures,
from 100°C to 550°C. This chart is available in the appendix of this paper for a better view.
As thefractionof the so-called bainitic ferrite goes up to nearly 40 %, the Y-axis can be cut in
seven pieces of 5 % in order to plot the TTT diagram. It is worth noticing that the 0 %
transfonnation line cannot be defined this way, so it will be omitted.
TTT diagram
6001
'
•
Log Time fsl
Figure 70: TTT diagram drawn on the basis of the calculatedfractions of phase transformed
described in the figure 69.
The diagram of the figure 70 is relatively regular except forthe holding temperature of SSO^C
where there is a peak. There is no clear boundary between the bainite region and the ferrite
71
4. Discussion
TTT diagram for the annealing at 750"C
region, but this is normal for low alloyed steel ^^^\ On the other hand, we can see quite well
that Ms is between 250°C and 300°C. The value of 290°C was calculated earlier with the
equation (17). As a matter of fact, the martensitic transformation is extremely fast so that it
occurs during the quench and the only thing that one can observe later is carbide precipitation,
which does not produce an important change in length l^^-^'J. Because of kinetics reasons, the
precipitation is more important at higher holding temperature, and it explains the slight
levelling from 100°C to 250°C.
Figure 71 and figure 72 show two camemberts as illustrations of the phase's disfribution
inside the sample at the end of the isothermal holding at 500°C and 200°C, according to the
calculation.
500 "C
Bainite
!35%
Figure 71: Distribudon ofthe phases
after the following treatment:
750^/
lOmin/ 500°C/ 15min, according to the
Ferrite
55%
calculation.
200 "C
IVIart
25%
Figure 72: Distribution of the phases
after the following treatment: 750"C/
Ferrite
55%
lOmin/ 200V/ IJmin, according to the
calculation.
Figure 71 : at 500°C, the phase that forms is bainite. According to the calculation, its amount
goes up to around 35 %, so that there remains 10 % of untransformed austenite, which is quite
a high volume fraction. The section 4.7 will come back to this part ofthe discussion. Figure
72 : at 200°C, we are between Ms and M f A linear relation between the fraction of martensite
and the temperature leads to think that there must be something like 25 % of martensite versus
20 % of austenite remaining. This austenite will finally transform to martensite during the
later cooling to the room temperature. Figure 73 shows a 3-dimensions TTT diagram based on
this data. The surface contains the iso-transformation curves. This chart was drawn with the
software Matlab. A bigger picture is available in the appendix.
72
4. Discussion
Decayed series
3D T T T diagram
Figure 73: Three-dimensions TTT diagram (the same as the one of figure 70).
4.4 Decayed series
As described in the section 4.2, the data corresponding to the intercritical annealings at SOOT
and 900°C were not relevant to plot TTT diagrams because of the parasite transformations
taking place during the quenches. On the other hand, these dilatometric data contain many
interesting information about the steel in a metallurgical point of view. The experiments were
not pointless.
4.4.1 Quenches from 900°C
1201—•
1140
•
1145
Change in length
•
—
1150
1155
Time [sl
1160
1165
Figure 74: Dilatation curves of three quenches carried outfrom 900°C
and called R910,
RR9andRR9B.
The quenches performed with the dilatometer present important inaccuracies. The user can be
victim either of a lack of gas pressure during the blowing or of a lack of heat transfer by
73
4. Discussion
Decayed series
convection from the gas to the sample. Nevertheless, the use of helium is an amelioration
compared to the nitrogen. In order to describe the difficulties that one meets when performing
quenches, the charts of the figure 74 represent three quenches from 900°C under different
aspects.
Temperature
•
10001—'
1140
1145
•
•
1150
1155
Time fsl
'
1
1160
1165
Figure 75: Cun>es showing the temperature of the sample for 3 quenches from POOV.
The first quench, called R910, was programmed at the cooling rate of 300 K/sec. Of course,
one should be aware that the real cooling rate could never be so high, this is why the program
put the quenching gas still available for 20 seconds more, which should be long enough for
the sample to join the programmed isothermal holding temperature.
Cooling rate
50
' I
A
1 - 50
5!
2- 100
O)
1-150
j^^^^^^^
RR9B
tr°
o
Ü
-200
RR9
-250
1140
1145
1150
1155
Time fsl
1160
1165
Figure 76: Instantly measured cooling rate for 3 quenches from 900 "C.
RR9 was programmed to quench at the rate of 150 K/sec, and it managed quite well, in
comparison with R910. Both transformations in R910 and RR9 are visible on figure 74 of
course, but also onfigure75. The heat released by the transformation induces a slight hump in
74
4. Discussion
Decayed series
Picture 11 (RR9): SEM micrograpii ofa sample austenitised at 900''Cfor 10 minutes, then
quenched to room temperature. It is fully martensitic. Grain size : ± 15 /um.
(-1
(UJ
I
-•-^i
Its.?
- • ^ ¥ r / f h'
n r
wm
-/}//
Picture 12 (R910): SEM micrograph ofa sample austenitised at 900''Cfor 10 minutes then
quenched to 100''C. Bainite or Widmanstatten ferrite.
75
4. Discussion
Decayed series
Picture 13 (R910): Optical micrograph, ofa sample austenitised at 900''Cfor 10 minutes then
quenched to 100''C. Acicular ferrite of Widmanstatten.
Figure 76 siiows the derivative of the curves showed in figure 75. It also gives an idea of the
maximum cooling rates that have been reached (more than 200 K/s).
The conclusion is that only the experiment "RR9" reaches Ms without having undergone any
other transformation. A better control during the quenches for the whole set of experiments
would have been possible i f hollow samples had been used. Indeed, hollow samples are
lighter, which means a smaller weight of material that must be cooled with the same gas blow.
Furthermore, it is possible to add a pipe in order to blow gas inside the sample. On the other
hand, only solid samples allow to perform microscopy afterwards, since hollow samples are
too thins.
Picture 11 presents the microstructure of sample RR9, while picture 12 and 13 present the
microstructure of sample R910. As expected, the first one contains 100 % of martensite, while
the microstructure of the second one is more complex. The sample R910 has been firstly
analysed with a scanning electron microscope (SEM) on the picture 12, and one could not say
if the microstructure was constituted of bainite or Widmanstatten ferrite. According to
H.K.D.H Bhadeshia, an optical microscope is more appropriate for the distinction between
76
4. Discussion
Decayed series
acicular ferrite and bainite. With an optical microscope, the bainite appears dark and the
acicular ferrite appears clear, as can be seen on picture 13
R910 is acicular ferrite.
4.4.2 Elements for C C T diagrams
If, for the intercritical annealing at 800°C and 900°C, the quenches are too slow for the
construction of TTT diagrams, the experiments can still be useful to collect data for CCT
diagrams. Indeed, the cooling rates were measured for each quench, as well as the
temperatures where some transformation begin. The following figures might help to
understand how measurements were done.
R820
1000
1078
1080
1082
1084 1086
Time fsl
1088
1090
Figure 77: Measurement qf the cooHng rates after 0.5 and 2 seconds,
and the average cooling rate.
R820
120
^100
^
f
80
•a
^
60
0)
? 40
ro
20
/
200
/
400
600
Temperature f°Cl
800
1000
Figure 78: Identification of the temperature of the transformation
that occurs during the quench.
11
4. Discussion
Decayed series
The example showed on figures 77 and 78 is a quench from 800°C down to 200°C. On the
first chart, one can see that the top and the end of the slope must be defined. Three values are
calculated on figure 77:
• the average cooling rate during the first half second. (135 IC/s in this case).
• the average cooling rate during the two first seconds. (81 K/s),
• the overall average cooling rate, (61 K/s).
The second chart (figure 78) presents the change in length as a function of the temperature,
which allows to identify the temperatures at which a transformation begins or ends. On this
example, there is the end of a "temperature gradient effect" around 740°C (as related in the
section 4.2.2), and the beginning of a transformation around 680''C. These measurements have
been applied to all the quenches that have been performed in this work.
The interest is, firstly, to see i f there are important variation for measurements that should be
identical, and secondly, to get the transformations temperatures, for instance for the
transformation that occurs during the quench from 800°C. The results for the quenches from
900°C are summarised in the table 9.
From - To
CR at 0.5 s
CR at 2 s
Average CR
First Transf.
900-80
900-100
196
107
145
95
63
17
440
660
900-100 bis
41
41
38
600
400
900-150
132
107
54
560
"410"
900-200
137
111
57
560
"400"
900-250
140
106
61
600
"380"
900-300
135
103
51
600
900-350
134
137
85
"450"
400
900-400
130
119
86
"500"
400
900-450
126
111
77
580
900-500
149
140
120
"740"
Sec. Transf.
Table 9: Syntixesis ofthe parameters measured on samples quenched from 900"C.
Except for the "900-100 bis", where the program was set at 40 K/s, and "900-80", which is an
exception, the average cooling rates after 0.5 and 2 seconds do not vary too much, whereas
the overall cooling rate is higher for shorter quenches, i.e. for quenches to higher holding
temperatures. As one may notice that the instantaneous cooling rate does not stop to decrease
78
4. Discussion
Decayed series
all along the quench, it is normal that the average cooling rate of a longer quench will be
slower.
RR9b
1201—
0
1
200
r
1
400
600
Temoerature f»Cl
1
1
800
1000
Figure 79: Dilatation curve ofa quencit carried out from 900''C at 40 K/s.
Concerning the temperatures of the main transformations, we can roughly identify 600°C and
400°C. In the table 9, the figures surrounded by quotation marks identify a less pronounced
transformation. According to Andrew's formula, 600°C corresponds to the bainite start, while
400°C corresponds to the martensite start. The quench at 40 K/s aUows to see the formation of
both bainite and niartensite. One can also notice that the slope of the curve is more important
before the first transformation than after the second one. (oCaustenite > oCmartensite)
From - To
CR at 0.5 s
CR at 2 s
Average CR
First Transf.
Sec. Transf.
800-80
800-100
132
99
93
72
58
44
"740"
"740"
700
700
800-150
128
88
67
"740"
700
800-200
135
81
61
"740"
680
800-250
114
74
61
"740"
700
800-300
103
71
64
"740"
700
800-350
129
94
84
"740"
700
800-400
138
91
86
"740"
700
800-450
101
71
67
"740"
715
800-500
142
97
92
"740"
680
Table 10: : Syntixesis of tiie parameters measured on samples quenciied from 800°C.
The main characteristics of the quenches described by the table 10 are the end of the plateau
around 740°C and the beginning of a transformation at already 700''C. At the beginning of the
quench, the sample contains about 10 % of ferrite and the carbon enrichment of the austenite
79
4. Discussion
Decayed series
is around 0.18 wt. %. This amount of carbon is still low and cannot hinder a diffusive phase
transformation, furthermore, the small ferrite phase (10 %) constitutes excellent nucleation
sites. This explains the fact that in the case of an annealing at 800°C, the austenite transforms
faster
than after a complete austenitisation at 900°C (transformation at 700°C instead of
óOO^C).
From - To
CR at 0.5 s
CR at 2 s
Average CR
First Transf.
Sec. Transf.
750-80
750-100
83
152
89
110
54
61
"725"
"725"
300
290
750-150
130
94
64
"725"
295
750-200
101
103
79
"725"
300
750-250
81
72
61
"725"
290
750-300
96
79
69
"725"
750-350
120
94
90
"725"
750-400
107
78
71
"725"
750-450
77
72
71
"725"
750-500
113
102
102
"725"
750-550
132
102
104
"725"
"680"
Table 11: Syntiiesis ofthe parameters measured on samples quenched from 750"C.
As for the 800°C series, the 750°C series, described on the Table 11, also show a pronounced
"gradient effect" plateau whose end is around 725°C. For tempering at temperatures lower
than 300°C, a fast phase transformation is observable near 290°C and it corresponds to the Ms
temperature of the austenite charged with 0.35 wt.% C. No transformation was experienced
above the martensite stait, and this is why the data coming from the 750°C series was kept for
the drawing of a TTT diagram.
Indeed, the reason why there was no parasite transformation is that the higher carbon content
has stabilised the austenite enough to push the transformation noses to the right (on a TTT
diagram)
In a general way, one can see that, for comparable experiments, the values of the cooling rates
differ quite much despite the fact that programmed cooling rate were identical. It is very
difficult to get two times the same results, in what concerns the quenches. This illustrates the
fact that even on a very good dilatometer, a quench is still something difficult to control,
however it must be easier with hollow samples.
80
4. Discussion
Remedy : formula applied upon the quench
4.5 Remedy: Formula applied upon the quench
This part sketches out a technique for the calculation of the phase fraction transformed during
a quench, in contrast to an isothermal tempering. In the equation (3) (— =
express
M=Y-YQ.
V -V
-), we can
In the case of the isothermal holding, the reference (7o) for the
calculation of the growth of the new phase was simply the change in length measured at the
end of the quench, i.e. at the holding temperature, and when no transformation had occurred
yet. This value could be kept constant as the holding was carried on in isothermal conditions.
The method can easily be adapted to a non-isothermal transformation, if the reference 7o is
properly dependent on the temperature. Hence in the coming calculation, Y will be the
measured data while YQ will be calculated making the assumption that the phase fraction
remains constant. Let us firstly look at the reference curve beside the data curve in the case of
a quenchfrom800°C to 350°C on figure 80.
Quench from QOO°C to 350''C
Time fsecl
Figure 80: Dilatation curves fi-om the beginning of the quench (8 OOV) to the end of the
isothermal holding (350V). Red: experiment, blue: calculated assuming that the austenite
does not trcmsform.
Thanks to the logarithmic X-axis, it is easy to distinguish the quench, that lasts about 10
seconds, and the isothermal holding that goes on for 15 minutes. In this case, the reference
curve YQ shows the change in length of a sample that would contain 90 % of austenite and 10
% of ferrite and that would not undergo any transformation. Something happens in the
microstructure when the Y curve moves away from YQ during the quench (around 1 second).
81
4. Discussion
Remedy : formula applied upon tlie quencit
and we can see that there is still a transformation going on during the tempering because the
change in length still increases.
R835
1.51
•
'
•
.
•
1
Figure 81: Fraction of bainitic ferrite calculated on tite basis ofthe results
showed on figm-e 80.
The values of AI = Y-YQ can now be used by the equation (6) or (9) in order to calculate the
amount of bainitic ferrite. The figure 81 presents the result obtained for the example
mentioned above. It indicates that the transformation starts after about one second, then it
grows very fast until the end of the quench and it finally goes on slowly. A problem is that the
bainitic ferrite fraction goes up to more than 100 %, which is impossible. That inaccuracy is
mainly due to a bad estimation of the expansion coefficient of the austenite. On the figure 80,
one can see that the calculated change in length at the temperature of 350°C is close to 0,
which seems to low with regard to the previously observed experiments.
From the beginning of this work, we have made the assumption that the expansion of the y
phase as a function of the temperature is linear, and so that oty values 23.5*10"'' K"'. Actually,
the expansion of the austenite is non-linear, and it is difficult to get good data for its
expansion coefficient at lower temperatures. One way to refine this parameter is to choose a
dilatometry experiment where we know that no transformation happened during the quench,
and to make fit the calculated values of the change in length to the measured data. Indeed, the
figure 82 shows the quench from 750°C to 350°C, The fit seems quite good in this case, as the
curves remain close to each other, but looking at the figure 83, which is the calculated fraction
of bainitic ferrite, we see that the growth of the ferrite phase is inverted when the YQ curve
82
4. Discussion
Remedy: formula applied upon the quench
happens to be above the 7 curve. It may be due to the fact that the experimental data are
falsified by the "plateau" effect. This imperfection adds to the difficulty ofthe calculation.
Quench from 7m°C to 350»C
100<
O
*
80-
x:
D)
C
0)
c
<B
O)
c
(Ö
.c
60 •
iY
40-
O
Vo
20 •
0^
10
Time fsecl
Figure 82: Dilatation curves from the beginning of the quench (VSOV) to the end of the
isothermal holding (350V). Red: experiment, blue: calculated assuming that the austenite
does not transform.
0.8|
•
•
•
'
10°
R735
—
0.61
-0.2'
10"^
•
^
10^
•
'
10"
Time fsecl
Figure 83: Calculatedfraction of bainitic ferrite on basis offigure 82.
As a final example, we may look at the figures 84 and 85, which show the result of this
calculation applied to the quench of a completely austenitised sample, i.e. from 900°C. Hence
there is a real problem with the accuracy of the evaluation of the expansion coefficient ofthe
austenite phase. Solving this problem is quite difficult since a good estimation of this
parameter would require new experiments, and in order to get a higher precision, the value
83
4. Discussion
Remedy: formula applied upon the quench
should be given by a polynomial equation taking the temperature into account to the third
degree.
Quench from 900°C to SSO^C
I
•
10"^
1
•
10°
1
.
10^
1
10"
Time fsecl
Figure 84:: Dilatation curves from the beginning of the quench (900V) to the end of the
isothermal holding (350V). Red: experiment, blue: calculated assuming that the austemte
does not transform.
R935
1.5
a)
1
bai
o
c 0.5
raction
»*o
0
.0.5i
10"'
.
^
10°
^
.
10'
^
1
io"
Time fsecl
Figure 85: Calculatedfraction of bainitic ferrite on basis offigure 84.
84
4. Discussion
Intercritical annealing at 750°C : test
4.6 Intercritical annealing at 750-C: test
4.6.1 Explanations for tlie gap between the annealings at 750"C
The gap is visible in figure 58. Thi-ee explanations are faced :
•
The first explanation for the deviation between the two dilatation curves was linked to the
difference in the phase fractions and to the fact that the austenite is more compact than the
ferrite. It has been detailed previously.
•
A second track is the possible experimental deviation. Indeed, the absolute measure for
the change in length is not always rehable : the height of two different dilatation curves
can sometimes differ, especially when the treatment contains a quench, which might move
the sample between the quartz rods (figure 7). But as there was no such operation for these
experiments before the end of the annealing, this second explanation can be ignored.
•
Finally, a deviation can be seen in the fact that one of the two samples has undergone a
holding at 950°C. As it has been austenitised, its pearlite has been decomposed and then it
comes back in a different way. As a matter of fact, the pearlite formed after the hot rolling
is not the same than the one that will be formed later, and thus the overall dilatation is
changed. This theory was proposed by Theo Kop, from the Materiaalkunde in the
TUDelft.
4.6.2 Microstructure
The microstructure showed on the picture 8 is very strange because of its strong
heterogeneity. Nobody could explain the reasons of that strange phenomenon. All the more, it
cannot be due to a mistake since the experiment was performed twice, and since the dilatation
data files prove the accuracy of the treatment.
After a long time of thinking, the beginning of an explanation exists. The figures 56 and 57
show us that coming from 100 % of y phase at high temperatures and stopping at 750°C, we
are really on the Ar3 point, which means that some ferrite should begin to appea:- in the
austenite matrix. Actually, at Ar3, the driving force for the formation of ferrite is very low,
thus very few nuclei of a phase succeed to reach the critic size. And, as the temperature is
quite high, the diffusion is such that the surviving nuclei are able to grow quite fast.
These words account for the partition of the microstructure in two : a fully austenitic region
and a ferritic region. Now let us say that the austenitic region will become, during the quench,
85
4. Discussion
Intercritical annealing at 750°C : test
the fully martensitic structure that one can see on the picture 10, The problem now is : why is
there a banded structure instead of huge ferrite grains ?
As a matter of fact, the assumption of the growth of only a few ferritic grains does not hold
since the picture 9 shows relatively smaU grains of ferrite beside grains of martensite (10pm).
Nevertheless, the banded structure is related to the non-homogene repaitition of the
manganese, when the metal is hot-rolled. As this element is in a substitutional position, it does
not move a lot during the next thermal treatments, and thus it confers a so-called memory to
the microstructure. Since the manganese is gammagene, austenite will appear preferably near
it, and as a consequence, ferrite will appeal' preferably away from it. Finally, the distribution
of the carbon through the microstructure is influenced by the distribution of the manganese,
and it explains the fact that the pearlite is aligned in the sample "^^^l
Furthermore, an important effect of this non-homogeneity is that the temperature A r l is
different from one place to another depending on the local concentration in the alloying
elements. This may have introduce different types of nucleation during the intercritical
holding at 750°C.
Unfortunately, it still does not give the key to the current problem.
86
4. Discussion
Amount of retained austenite
4.7 Amount of retained austenite
Since this research is partly an investigation on the possibilities of producing a low silicon
TRIP-aided multiphase steel, it was interesting to observe the samples heat treated with the
dilatometer and to look for possible retained austenite, Among the whole set of heat
treatments, figure 86 recalls those that were applied for the study of multiphase steels.
Treatments
Time fmin]
Figure 86: Tixermal treatments for tiie drawing of tiiree TTT diagrams.
Three annealing temperatures were faced (750°C, 800°C, 900°C), and the duration of the
annealings was of 10 minutes. Concerning the holdings, they were spread from 100°C to
500°C by steps of 50°C, and were lasting 15 minutes. Which of these thermal treatments are
the most susceptible to give stabilised austenite ? As already written in the introduction
(1,2,2), the stabihsation of the austenite is the resuh of a two-steps carbon enrichments : the
first one during an intercritical annealing, and the second one during the formation of bainite,
ideally on the second isothermal holding. Ever since, between the two annealing temperatures,
the one that provides the best carbon enrichment is 750°C, as observed in the section 3.2.3.
Then, for this series, which one of the holding temperature was the best ? Since the Ms
temperature was calculated around 290°C for an austenite enriched at 0.35 wt. % C, and Bs
was calculated at 600°C (with Andrew's formulas (16) and (17)^^^^), aU the levels comprised
between Ms and Bs are worth to be analysed more closely.
87
4. Discussion
Amount of retained austenite
_ ,
—
-
—
Picture 14 (R745): SEM micrograpii ofa sample annealed at JSO^Cfor 10 minutes,
quenciied to 450"C, then held for 15 minutes.
Picture 15 (R750); SEM micrograph ofa sample annealed at 750 °Cfor 10 minutes,
quenched to 500°C, then held for 15 minutes.
88
4. Discussion
Amount of retained austenite
The samples corresponding to those treatments were thus analysed by scanning electron
microscopy and X-ray diffraction (performed by Lie Zhao, from the TUDelft). The result of
the microscopy analysis is that no stabilised austenite could be observed with the SEM, and
the results of the measurements performed by X-rays prove to be very low. As a matter of
fact, there was not enough silicon to impeach carbides precipitation in the austenite. The
pictures 14 and 15 show the microstructures of samples annealed at 750°C and quenched
respectively to 450°C and 500''C.
One can see on both pictures 14 and 15 that the grain surrounded by ferrite is decomposed
bainite. hideed, the clear lines that are crossing the grain are carbides that were precipitated on
the sides of the ferritic laths during the bainitic transformation
Retained austenite cannot
be observed in these microstractures, neither by the SEM, nor by X-ray diffraction. Maybe
this austenite is too thin to be detected by XRD. For high-silicon steels, large amounts of
austenite had been obtained for a similar heat treatment : intercritical annealing at 750°C
foUowed by a bainitic holding at 400-450''C for several minutes ^^^'^^l
One can think that an unavoidable condition for stabilising austenite is to add a graphitising
element such as silicon or aluminium in order to prevent cementite precipitation. This is not
true and Pascal Jacques, from the Université Catholique de Louvain, has proved it. He has
shown that it was possible to stabihse austenite in a low-sihcon steel and to obtain the TRIP
effect
For the composition : 0.18 wt.% C, 0.39 wt.% Si and 1.33 wt.% Mn, and by
applying the following heat treatment; 730°C/5 min/370''C/l min, he could get up to 8.5 % of
retained austenite. Furthermore, what influences the properties of a TRIP-aided steel is not
only the amount of austenite that it contains, but also the stability of this phase. And as a
matter of fact, the mechanical properties of the low-silicon TRIP-aided multiphase steel that
was produced avowed to be excellent
What explains the very low concentrations of retained austenite measured in the samples
produced in the present work is the length of the isothermal holding. A bainitic holding of
15 minutes is too long (for a low silicon steel) since it aUows most of the austenite to
transform by carbides precipitation ^^l (The Mössbauer spectroscopy measurements of Pascal
Jacques give an amount of 2 % of retained austenite for a holding of 10 minutes ^^l) The
austenite retention in low-silicon steels is only possible i f the kinetics of the bainitic
transformation are taken into account. The bainitic holding must be long enough for the
carbon to diffuse out of the supersaturated ferritic laths into the surrounding austenite, but it
should be stopped before the transformation of this austenite.
89
4. Discussion
Amount of retained austenite
But one should keep in mind that one of the aims of the present work was to produce TTT
diagrams, which explains the need to apply sufficiently long holding times, finally :
15 minutes. The samples could not therefore have undergone the ideal heat treatment for the
production of TRIP-aided steels.
90
5. Conclusions
5. Conclusions
This study investigated the heat treatments for the production of a low-siUcon low-carbon
TRIP-aided muhiphase steel by means of dilatometry. Programs have been written for the
interpretation of the dilatometric curves of, firstly the transformation a-^y upon heating, and
secondly, the isothermal transformation during bainitic holding. The output of this second
program have been used to release a TTT diagram for the phase transformations following an
intercritical annealing at 750°C. For samples annealed at 800°C and 900°C, information on the
transfomations that occurred during the quench have been reported. It may be useful to draw
CCT diagrams, for instance.
Moreover, a method for the measurement of the phase transfonnation during the quenches has
been developed (Section 4,5). Besides the research for the TTT diagram, an experiment about
the intercritical annealing has been canied out (Section 3.4). The observed microstmcture still
constitutes a mystery.
One should be aware that the calculation method developed in this thesis work require very
accurate data for the different pai'ameters that describe the material. Moreover, the calculated
phase fractions are very dependent on the input data such as the lattice parameters and the
expansion coefficients (Section 4.2.4). Furthermore, perfect quenches are needed for the
calculation of TTT diagrams: the use of hollow samples are therefore recommended for the
dilatometry tests.
The perspectives for the possible continuation of this work could be :
• to get some new dilatometric data, but with hollow samples this time.
• to try and refine the parameters for the calculations.
• to integrate the programs in a more efficient programmatic language.
For what concerns the steel grade that has been studied in this work, mechanical tests of this
material are being cai-ried out currently at the university of Aachen, Germany. Furthermore,
the possibilities of production of a TRP-aided multiphase steel of this composition have been
investigated by Hoogovens. Laboratory simulations of a hot-dip galvanizing hne have proved
to give a satisfying result ^^^l
91
5. Conclusions
References
[1]
H, Shirasawa, "High-Strengtit steels for automative, symposium proceedings";
Slater, Baltimore, M D ,1994, pp 3-10.
[2]
I . Tamura, Metal Sc., 1982, vol 16, pp 245-252.
[3]
Anil K, Sachdev, Acta Metall., 1983, vol 31, n° 12, pp 2037-2042.
[4]
P. Jacques, PhD Thesis, UCL, Belgium, 1999.
[5]
P. Jacques, X. Cornet, Ph. Harlet, J. Ladrière, F. Delannay, "Eniiancement of tite
Mecitanical Properties of a Low-Carbon, Low-Silicon Steel by Formation of a
Multipitased Microstructure Containing Retained Austenite", Metall. Mater. Trans. A,
1998, vol 29A, pp 2383-2393.
[6]
H. Koh, S. Lee, S. Park, S. Choi, S. Kwon, N. Kim, Scripta Mater., 1998, vol 38(5), pp
763-769.
[7]
H. K, D. H. Bhadeshia, D.V. Edmonds, MetaU. Trans. A, 1979, vol lOA, pp 895-907
[8]
M . Takahashi, H.K.D.H. Bhadeshia, Mater. Trans. JIM, 1991, vol 32, pp 689-695.
[9]
H.K.D.H. Bhadeshia, "Bainite in steels". The institute of Materials, London, 1992, pp
72-74.
[10]
O. Matsumai-a, Y. Sakuma, Y. Ishii, J. Zhao: Iron Steel Inst. Jpn. Int., 1992 vol 32
(10), pp. 1110-16.
[11]
Anne Mertens, Project for the FRIA, 1997.
[12]
G.R. Speich: "Fundamentals of Dual-Phase Steels", R.A. Kot and B.L. Bramfit, eds..
Trans. Met. Soc. AIME, Warrendale, PA, 1981, pp 3-45.
[13]
Jacobien Vrieze, Walter Vortrefflich, Laurens de Winter, Hoogovens Research report,
1999.
[14]
Bhadeshia H.K.D.H, "Thermodynamic
analysis of
diagrams", Metal Science, 1982, vol. 16, pp 159-165.
http://Engm01.ms..ornl.gov/TTTCCTPlots.html
[15]
F. Delannay, "Compliments de Metallurgie Piiysique ", Cours MAPR 2420, UCL.
[16]
E. Girault, P. Jacques, K. Mois, P. Harlet, J. Van Humbeeck, E. Aernoudt, F.
Delannay: "Material Ciiaracterisation", 1998, vol. 40(2), pp 111-118.
92
isotiiermal
CE.
transfonnation
[17]
L. Zhao, T. Kop, J. Sietsma, S. van der Zwaag, "Dilatometric analysis of bainitic
transformation in TRIP steels", 1999, to be pubhshed in Euromat '99.
[18]
K.W. Andrews, J. h-on Steel Mst., 1965, pp 721-727.
[19]
Caian Qiu, Sybrand van der Zwaag, "Dilatation measurements of plain carbon steels
and tiieir tiiermodynamic analysis", Steel Research, 1997, issue 1, pp 32-38.
[20]
D.J. Dyson and B. Holmes, J. t o n and Steel Institute, May 1970, pp 469-474.
[21]
N . Ridley, H. Stuart, L. Zwell, "Lattice Parameters of Fe-C Austenites at Room
Temperature", Trans. Met. Soc. AIME, August 1969, vol 245, pp 1834-1836.
[22]
K. H. Jack, "Structural Transformations
Martensitic Steels", Sept. 1951, pp 26-36,
[23]
J, Gordine, I , Codd, "Tiie influence of silicon up to 1.5 wt % on tiie tempering
cliaracteristics ofa spring steel", J,of the Iron Steel Inst., April 1969, pp 461-467.
[24]
R.W.K. Honeycombe, H.K.D.H. Bhadeshia, "Steels", MMS, Edwai'd Arnold, 1995
(Sec. Ed,), pp 145-146.
[25]
A. A l i , M . Ahmed, F.H. Hashmi, A.Q. Khan, Metall. Trans. A, 1993, vol 24A, pp
2145-2150.
[26]
Y. Sakuma, D.K. Matlock, G. Krauss, Metall. Trans. A, 1992, vol 23A, pp 1221-1232.
93
in tiie Tempering
of
Higii-Carbon
6. Appendix
6. Appendix
6.1 Guidelines/Manual bainitic transformation analysis program
Introduction
This program calculates from a dilatometric data file the fraction of bainite that forms during
the holding at the transformation temperature. It uses the lattice parameters of austenite,
ferrite and cementite as well as their expansion coefficients. The calculation involves the
solvation of an equation by the iteration method of Newton-Raphson. Two cases are taken
into account : the quench is done from a complete austenisation temperature or from an
intercritical temperature. The user is asked for the annealing temperature, and in the case it is
intercritical, he must give the fraction of austenite that was in the sample before the quench.
The user also has to introduce the holding temperature at which the isothermal transformation
is meant to occur.
Matlab environment
To run this program, you need a Matlab software on your computer, In the main directory
named ''matlab", a sub-directory (for instance -"bain") should be created in order to host the
program files and the data files. This is also the place where the output files will be saved. At
the beginning, the directory ''bain" should contain the following files which are the main
program and its sub-routines : bainite.m,fprim.m,
gtf.m.
Pre-processing
Make the input file. This is a table in wri format with 4 columns : anvthing(number of the
value), time(s), temperature(°C), dilatation change r*10^V The table must start at the very first
hne of the document, otherwise Matlab would not recocknize it as a matrix. Don't forget to
put this wri file in the directory "bain".
Processing
hi the Matlab main command window, type : cd bain, then "enter", and bainite followed by
"enter". After the program has started, you will be asked for the name of the input file. I f it is
for instance data.wri, you must type : 'data' with the quotation marks.
94
6. Appendix
Then you have to enter the name of the output file according to the same typing rules, for
instance : 'output' and the file output.wri wiU be created at the end of the calculation and the
results will be stored inside.
Finally, you are asked for the annealing and the holding temperatures of the experiment. In
the case it is intercritical, you have to type the fraction of austenite present in the sample at the
end of the annealing.
Post-processing
The program has released two windows ; the first one contains a graph of the retained
austenite as a function of the time, and the second one is the amount of bainite as a function of
the time.
The output file is a table of three columns : time(s), fraction of retained austenite, fraction of
bainite. It can be converted then to an excell file or used as is by an other Matlab program.
The interest of using Matlab is that the user can easily make modifications to the program, for
instance: the lattice parameters. For details, see the Matlab Primer.
6.2
dispC
Programs
BAINITIC TRANSFORMATION');
% BAINITE figures out the fraction of retained austenite that appears
%
during the bainitic transformation of a low silicon steel
%
quenched from an intercritical temperature. The data needed
%
is a four columns text file that contains the time in the second
%
column, the temperature in the third and the change In length
%
in the fourth. BAINITE uses the Newton-Raphson method on the
%
function GRF, which has been written by Lie Zhao.
%
The output is a .wri file that contains three columns. The first
%
one is for the time, the second one is for the austenite fraction
%
and the third one is for the bainite fraction.
global T, global DL, global FGI, global DLO;
name = input('Name of the input file .wri ? ');
eval(['load ',name,'.wri']);
atemp = input('Annealing temperature ? ');
If atemp == 900,
FGI = 1;
95
6. Appendix
else
FGI = input('Fraction of austenite at the end of the annealing : ');
end;
htemp = input('Holdlng temperature ? ');
[r c]=size(eval(name));
Time = eval([name,'(:,2)']);
Temp = eval([name,'(:,3)']);
deltal = eval([name,'(:,4)']);
fgamma = zeros(r,1);
bain = zeros(r,1);
Time2 = zeros(r,1);
counti = debut(name);
%
%
The function "debut" finds the beginning of the
isothermal holding, which defines DLO.
DLO = deltal(countl);
timeO = Time(countl);
for count2 = counti :r,
T = Temp(count2);
DL = deltal(count2);
fgb=0.5; twin=0;
while (abs(fgb-twin)>0.0001),
twin = fgb;
fgb = fgb - (grf(fgb))/(fprim('grf',fgb));
if abs(fgb-twin)>1000, error('No convergence,'), end
end;
fgamma(count2) = fgb;
Time2(count2) = Time(count2) - timeO;
if
((Time(count2)<Time(count1)+900)+((Time(count2)>=Time(count1)+900)*(Temp(cou
nt2)<htemp-5))==0), break, end
end;
%
%
%
The "for" loop Is needed to go through the data. And the
"while" loop is for the refining of the convergence in
the Newton-Raphson method.
bain = - (fgamma - FGI);
figure(l);
clf;
semllogx(Time2(count1 :count2),fgamma(count1 :count2),'w');
title([num2str(atemp),'SG to ',num2str(htemp),'2C']);
xlabel('Time [sec]');
ylabel('Fraction of retained austenite');
figure(2);
96
6. Appendix
Clf;
semilogx(Time2(count1 :count2),bain(count1 :count2),'w');
title([num2str(atemp);öC to ',num2str(htemp),'^C']);
xlabel('Time [sec]');
ylabel('Fractlon of bainitic ferrite');
output = [Time2(count1 :count2),fgamma(count1 :count2),bain(count1 :count2)];
out = output';
fid = fopen([name,'op.wri -ascii'],'w');
fprintf(fid,'%4.4f\t %4.4f\t %4.4f\n',out);
fclose(fid);
function deb=debut(name)
% DEBUT prend comme argument le nom d'un fichier
%
de donnees dilatometriques. II renvoie
%
alors I'indice de la ligne qui marque le debut de
%
la transformation bainitique.
dat = [1090 1070 1025 1060 1030 910 1020 1025 870 858; 1200 1065 1075 1065
1040 910 1002 1020 861 0;1250 1157 900 1085 1125 870 995 1020 840 0];
asde = abs(name);
ro = asde(2) - 54;
te = 10*(asde(3)-48)+(asde(4)-48);
CO = (te-5)/5;
deb = dat(ro,co);
function xprim = deriv(x)
[r c]=size(x);
xprim = zeros(r,2);
forcount = 4:1:r-3,
xprim(count,1) = x(count,2);
xprlm(count,2) = (x(count+3,3)+4*x(count+2,3)+5*x(count+1,3)-5*x(count-1,3)4*x(count-2,3)-x(count-3,3))/(x(count+3,2)+4*x(count+2,2)+5*x(count+1,2)-5*x(count1,2)-4*x(count-2,2)-x(count-3,2));
end;
function fpr = grfint(fgb)
% GRFINT is a function used by the programme BAINITE.
%
The root of this function gives the value of the volume
%
fraction of retained austenite. This function is inspired
%
from the equation (13).
global T, global DL, global FGI, global DLO;
x i =0.16;
97
6. Appendix
X2 = 1.5;
aalpha = 2.883*(1+13.16-6*1+0.0037476-6*1/^2);
agammab = (3.6008+x1/fgb*0,046+x2*.00103)*(1 +23.5e-6*(T-25))agammai = (3.6008+x1/FGI*0.046+x2*.00103)*(1+23.5e-6*(T-25));
fpr = (fgb*(agammab/^3-2*aalpha/^3)+FGI*(2*aalpha/^3agammai'^3))/(3*(FGI*agammai/^3+2*(l-FGI)*aalpha/^3))-((DL-DL0)/10000);
6.3 Three-dimensions
TTT diagram
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