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23.- 25.5.2006, Hradec nad Moravicí
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A CONTRIBUTION TO THE MODELLING OF THE AUSTENITISATION
PROCESS IN STEELS
D. San Martína
F. G. Caballerob
C. Capdevilab
C. García de Andrésb
a
Fundamental of Advance Materials Group, Faculty of Aerospace Engineering, TU Delft,
Kluyverweg 1, 2629 HS Delft, The Netherlands, [email protected]
b
Solid-Solid Phase Transformation Group (MATERALIA), Department of Physical
Metallurgy, Centro Nacional de Investigaciones Metalúrgicas (CENIM-CSIC), Avda.
Gregorio del Amo, 8, 28040 Madrid, Spain, [email protected]
Abstract
Lots of industrial processes involved in the fabrication of steel components rely on heat
treatments which cause the steel to revert to the austenitic condition. The initial condition of
the austenite determines the development of the final microstructure and mechanical
properties, so the modeling of the non-isothermal formation of austenite is useful. In this
sense, a quantitative theory dealing with the nucleation and growth of austenite from a variety
of initial microstructural conditions is vital. The main aim of this work is to study the
mechanisms that control the austenitization process in steels with a ferrite and pearlite mixed
initial microstructure. The compiled knowledge in literature regarding the isothermal
formation of austenite from different initial microstructures (pure and mixed microstructures),
has been used in this work to develop a model for non-isothermal austenite formation in steels
with initial microstructure consisting of ferrite and pearlite. The microstructural parameters
that affect the nucleation and growth kinetics of austenite, and the influence of the heating
rate have been considered in the modeling.
1. INTRODUCTION
The formation of austenite during heating differs in many ways from those transformations
that occur during the cooling of austenite. For instance, the kinetics of austenite
decomposition can be completely described in terms of the chemical composition and the
austenite grain size. By contrast, the microstructure from which austenite may form is more
complex and additional variables are therefore needed to describe the kinetics of austenite
formation. Factors such as particle size, distribution and chemistry of individual phases,
homogeneity and the presence of non-metallic inclusions should all be important [1-4].
The development of dual-phase steels by partial austenitization revived the interest for the
heating part of the heat treatment cycle in the eighties[5-7]. Speich et al.[2] and, Garcia and
DeArdo [1] described in detail the mechanisms that control the austenite formation process
under isothermal conditions in low carbon steels with a ferrite-pearlite initial microstructure.
Later, Roosz et al. [8] quantitatively determined the influence of the initial microstructure on
the nucleation rate and grain growth of austenite during isothermal treatment of a eutectoid
plain carbon steel.
However, little information is available about the austenite formation in steels subjected to
continuous heating. In this sense, the aim of this work is to evaluate the influence of heating
rate and microstructural parameters on the non-isothermal kinetics of austenite formation in
steels with a ferrite and pearlite mixed initial microstructure.
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Table 1 Chemical composition (mass %)
Steel
C
Mn
Si
N
Al
P
Cr
Ni
Nb
MIXT
0.11
1.47
0.27
0.005
0.039
0.015
0.03
0.03
0.03
2. MATERIALS AND EXPERIMENTAL PROCEDURE
MIXT steel (Table I) is a low carbon manganese steel with a ferrite plus pearlite initial
microstructure. This steel was used to validate the kinetics model for austenite formation in
steel with a mixed initial microstructure. The material was soaked at 1473 K for 10 min., hot
rolled to 30 mm in several passes, and finally air cooled to room temperature. The as-rolled
microstructure of this steel formed by 87 % ferrite and 13 % pearlite is shown in Fig. 1.a.
Figure 1: Initial microstructure of MIXT steel: (a) Optical micrograph; (b) Scanning electron
micrograph.
Specimens were polished in the usual way and finished on 0.5 µm diamond paste for
metallographic examination. Two types of etching solution were used: Nital-2pct to reveal
microstructures by light optical microscopy and solution of picric acid in isopropyl alcohol
with several drops of Vilella’s reagent to disclose pearlite morphology on a JEOL JXA-820
scanning electron microscope. Figure 1.b shows a scanning micrograph of the morphology of
pearlite in MIXT steel.
Two parameters, the mean true interlamellar spacing, σo, and the area per unit volume of the
pearlite colonies interface, S vPP , characterize the morphology of pearlite [8]. The value of σo
in MIXT steel was derived from electron micrographs according to Underwood’s intersection
procedure [9, 10]. The value of S vPP was measured on scanning micrographs by counting the
number of intersections of the pearlite colony boundaries with a circular test grid as reported
by Roosz et al. [8]. Approximating the pearlite colony by a truncated octahedron, the edge
length of the pearlite colonies, a P , is calculated from the area per unit volume S vPP with the
following expression [11]:
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S
PP
v
(
)
6 1+ 2 3
=
8 2 (a
P
(a
)3
P
)2
=
(
31+ 2 3
4 2a P
)
(1)
Data for σo, S vPP and a P are listed in Table 2.
Table 2 Morphological characterization of pearlite in MIXT steel
Specimen
σo × 10-3, mm
S vPP , mm-1
aP × 10-3, mm
MIXT
0.15±0.02
1289±323
2.06±0.49
The experimental validation of the austenite formation kinetics developed in this work was
carried out using an Adamel Lhomargy DT1000 high-resolution dilatometer. In order to
analyze the progress of austenite formation in MIXT steel, interrupted heating experiments
were carried out by quenching. Heating dilatometric curves were analyzed to determine the
start temperature (Ac1) and the end temperature (Ac3) of austenite formation and then several
quench-out temperatures were selected in order to evaluate the progress of the transformation.
Austenite, which is formed during heating, transforms to martensite during quenching. Thus,
the progress of austenitization is determined throughout the evolution of the volume fraction
of martensite. Specimens from interrupted heating experiments were polished in the usual
way for metallographic examination. LePera’s reagent [12] was used to reveal martensite
formed during quenching. The quantitative measurement of martensite volume fraction was
carried out by point-counting method [13].
Figure 2. Optical micrographs from interrupted heating samples at different stages of the
reaustenitization process in MIXT steel. (a) 1017 K; (b) 1084 K; (c) 1135 K; Heating rate
of 5 K/s.
Figure 2 shows microscopic evidences of how austenite formation occurs in MIXT steel for a
heating rate of 5 Ks-1 throughout micrographs from interrupted heating samples at different
stages of the reaustenitization process. LePera’s reagent [12] reveals pearlite and ferrite as a
darker phase in the microstructure, whereas martensite formed during quenching appears as
lighter regions in the micrographs. Microstructure in Fig. 2.a is formed mainly of ferrite,
pearlite and some grains of martensite. At this quench-out temperature, the pearlite-toaustenite transformation starts. Figures 2.b and 2.c show intermediate stages of the reaction.
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3. RESULTS AND DISCUSSION
3.1
Modeling of Kinetics of Non-Isothermal Austenite Formation in a Steel with a
Ferrite Plus Pearlite Initial Microstructure
In the austenitization of microstructures composed of ferrite and pearlite, two different
transformations are involved: pearlite dissolution and ferrite-to-austenite transformation. Both
transformations take place by nucleation and growth processes.
Modeling of Kinetics of Dissolution of Pearlite.
Nucleation and growth processes under isothermal condition can be described in general
using the Avrami's equation [14]:
⎛ π •
⎞
Vγ = 1 − exp⎜ − N G 3t 4 ⎟
⎝ 3
⎠
(2)
•
where Vγ represents the formed austenite volume fraction, N is the nucleation rate, G is the
growth rate and t is the time. According to Christian [15], with a spherical configuration, an
•
exponent of 4 in time (t) in Avrami’s equation means that the nucleation rate ( N ) and the
growth rate (G) are constant in time.
•
Roosz et al. [8] proposed a temperature and morphology dependence of N and G as a
function of the reciprocal value of overheating (∆T = T-Ac1) as follows:
•
⎛ − QN ⎞
N = f N exp⎜
⎟
⎝ k∆T ⎠
(3)
⎛ − QG ⎞
G = f G exp⎜
⎟
⎝ k∆T ⎠
(4)
where QN and QG are the activation energies of nucleation and growth, respectively, k is
Boltzmann’s constant, and fN and fG are the functions representing the influence of the
morphology of pearlite and the heating rate on the nucleation and growth rates, respectively.
Several authors [8,16,17] reported that the nucleation of austenite inside pearlite takes place
preferentially at the points of intersection of cementite with the edges of the pearlite colony.
Approximating the pearlite colony as a truncated octahedron, the number of nucleation sites
1
per unit volume is calculated as N C ≈ P 2
where aP is the edge length of the pearlite
(a ) σ o
colony and σo is the interlamellar spacing [11].
Bearing in mind that the rate of nucleation increases as the pearlite interlamellar spacing
decreases and the edge length of the pearlite colony increases [18], and considering that the
•
heating rate ( T ) might influence on the nucleation rate, the function fN in equation (3) is
assumed to have the following general form:
fN = KN
(a )
P n
σ om
p
•
⎛⎜ T ⎞⎟ ( N ) r T
C
⎝ ⎠
•
(5)
where KN, n, m, p and r are empirical parameters. These parameters were adjusted in order to
obtain good fit between theory and the experimental austenite volume fraction curves. In this
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sense, the measured values of austenite volume fraction as a function of temperature can be
1
1
best described with n=6, m=1, p= and r= .
2
3
Austenite nuclei in pearlite grow when carbon atoms are transported by diffusion to the
ferrite/austenite boundary from the austenite/cementite boundary through the austenite and
from the ferrite/cementite boundary through the ferrite, resulting in a transformation of the
ferrite lattice to an austenite lattice [19]. As in the case of the reverse transformation
(austenite-to-pearlite transformation), the growth rate of austenite is believed to be controlled
by either volume diffusion of carbon or by boundary diffusion of substitutional alloying
elements [20,21,22]. If the growth rate of austenite is controlled by the bulk diffusion of
atoms in austenite ahead of the interface, the diffusion of carbon may play a more important
role than that of substitutional alloying elements. Diffusivity of the substitutional alloying
elements in austenite is far smaller than that of carbon. As a result, the substitutional alloying
elements may not diffuse a long distance during the reaction. However, as described by Porter
[23], when temperature decreases, boundary diffusion of substitutional alloying elements is
the dominant mechanism in the diffusion process. In that case, the partitioning of the
substitutional alloying elements is substantial during the growth of austenite and boundary
diffusion of the alloying elements may control the growth rate of pearlite.
The function fG in equation (4) representing the morphology dependence on the growth rate
can be expressed as follows:
fG = KG
1
(6)
σ oi
where KG is an empirical constant, i=1 if the growth rate of austenite is controlled by volume
diffusion of carbon and i=2 if the growth rate of austenite is controlled by boundary diffusion
of substitutional alloying elements [8].
A general equation to describe the non-isothermal overall pearlite-to-austenite transformation
in pearlitic steel was derived integrating the Avrami's equation over the whole temperature
range where the transformation takes place [24]. In this sense, we have taken logarithms and
∆T
differentiated in equation (2). Expressing time as t = • leads to:
T
dVγ
1 − Vγ
=
4π • 3 ∆T 3
NG
dT
4
•
3
⎛⎜ T ⎞⎟
⎝ ⎠
[
(7)
]
Integrating in 0, Vγ and [ Ac1 , T ] intervals on the left and on the right sides of equation (7),
respectively, it can be concluded that:
⎛
⎞
⎜ T
⎟
•
π
4
3
3
⎟
∆
Vγ = 1 − exp⎜ - ∫
N
G
T
dT
⎜ Ac ⎛ • ⎞ 4
⎟
⎜ 1 3⎜ T ⎟
⎟
⎝ ⎠
⎝
⎠
(8)
It has been assumed that at a heating rate higher than 0.5 Ks-1 the growth rate of austenite
would be mainly controlled by the volume diffusion of carbon in austenite, due to the fact that
the transformation would take place mostly at higher temperatures. Consequently, a i value of
1 is considered in equation (6) for that case. On the contrary, at heating rates lower than or
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equal to 0.5 Ks-1 the growth rate of austenite has been assumed to be controlled by boundary
diffusion of substitutional alloying elements and a i value of 2 is considered in equation (6)
for that case. The eutectoid temperature Ac1 of the steel was obtained using Andrews’ formula
[25].
Therefore, the austenite volume fraction obtained from pearlite dissolution, VγP , during
continuous heating of a ferrite plus pearlite initial microstructure is expressed as follows:
VγP
⎧
⎛
⎞⎫
⎜ T
⎟⎪
⎪
4π • 3 3 ⎟⎪
⎪
⎜
= V Po ⎨1 − exp⎜ - ∫
N G ∆T dT ⎟⎬
• 4
⎪
⎜⎜ Ac1 3⎛⎜ T ⎞⎟
⎟⎟⎪
⎪⎩
⎝
⎠
⎝
⎠⎪⎭
(9)
where VPo is the volume fraction of pearlite present in the initial microstructure. VPo = 0.13 in
MIXT steel.
Modeling of Kinetics of Ferrite-to-Austenite Transformation after Dissolution of Pearlite.
The purpose of this section is to develop a model describing the transformation process of a
mixture of carbon enriched austenite (obtained from the dissolution of pearlite) and ferrite,
during continuous heating. In the present model, the following assumptions are made:
-Ferrite-to-austenite transformation begins once the dissolution of pearlite is completed
(temperature Ac P ). No nucleation occurs during this transformation. All the nuclei are
generated during the dissolution of pearlite. Only growth of the existing austenite grains takes
place.
-The process is controlled by the volume diffusion of carbon in austenite. Diffusion of carbon
in ferrite has been ignored since the solubility of carbon in ferrite is negligible compared to its
solubility in austenite.
-The carbon content in the austenite after the dissolution of pearlite is assumed to have a mean
value of 0.8 wt-%. The carbon inside the austenite soon will diffuse to the austenite/ferrite
(γ/α) interface in order to re-establish the thermodynamic equilibrium at the corresponding
temperature.
-A planar geometry is considered in the modeling and local equilibrium is assumed. The mean
carbon composition of austenite grains is given by the equilibrium carbon concentrations
corresponding to the (α+γ)/γ phase boundary of the equilibrium diagram of the steel.
-The continuous heating curve will be expressed as a series of small isothermal steps
occurring at a successively higher temperatures with a time interval, dt , associated with each
step.
-At each step of the heating curve (i.e., each temperature), the time interval dt will be short
enough to make possible the assumption of steady state diffusion. Therefore, the gradient of
carbon in austenite is considered to be constant at each temperature. Finally, the diffusivity of
carbon in austenite is assumed to be only dependent on the local concentration of carbon in
austenite (different for each step of the heating cycle).
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Figure 3. Evolution of the gradient of carbon during ferrite-to-austenite transformation
( T ≥ Ac P ).
During the continuous heating, carbon atoms diffuse from inside the enriched austenite grains
to the γ/α interface, crossing it. Thus, the interface moves forward and some ferrite is
transformed into austenite. This process continues until the average carbon content in
austenite equals the carbon content of the steel, just when all ferrite has transformed
( Ac3 temperature) and the microstructure is fully austenitic. Under the above conditions, at
each step of the heating cycle, the diffusion equation for carbon in austenite can be simplified
to:
DCγ
d 2C γ
=0
dr 2
(10)
where DCγ is the diffusivity of carbon in austenite, C γ is the concentration of carbon in
austenite and r is the spatial variable normal to the interface γ/α . To solve equation (10), we
need to know the boundary conditions of the problem. As we stated before, we assumed the
existence of a gradient of carbon during the transformation from the center of the austenite
grains to the γ/α interface (Fig. 3). Therefore, for temperatures higher than Ac P ,
r = −ro ⇒ C γ = C c
(11)
r = r γα ⇒ C γ = C γα
(12)
where, C c and C γα are the carbon concentration in the center and at the interface of the
austenite grains, respectively. ro is the radius of carbon enriched austenite grains after the
dissolution of pearlite and r γα is the position of the interface γ/α. Solving (10) for the
boundary conditions (11) and (12), gives the equation for the gradient of carbon inside
austenite grains at each temperature,
(
dC γ
C c − C γα
=−
dr
Lγ
)
(13)
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The effective diffusion distance of carbon in austenite, Lγ , can be deduced from mass carbon
conservation. The total carbon concentration of the system at T = Ac P must be equal to the
carbon concentration at any T > Ac P . Applying this principle we get,
Lγ =
(
2r γα C eut − C αγ
(2C
eut
−C −C
c
)
)
(14)
γα
With C eut the eutectoid carbon composition (0.8 wt-%) and C αγ the carbon composition at
the interface inside the ferrite. On the other hand, the conservation of the carbon concentration
fluxes at the γ/α interface at each time interval is given by,
(C γα − C αγ )dr γα = − D γ dCdr
γ
(15)
dt
C
r =r
γα
Replacing equation (14) in (13) and the resultant in (15) gives,
(
)(
dr γα DCγ C c − C γα 2C eut − C c − C γα
=
dt
2r γα C γα − C αγ C eut − C αγ
(
)(
)
)
(16)
Integrating (16), an expression for the radius of the austenite grain during the transformation
is reached,
r γα = ηt 1 2
(17)
where η is the parabolic growth rate constant of the transformation,
(
(
⎡
C c − C γα
η = ⎢ DCγ γα
C − C αγ
⎣⎢
) (2C
) (C
)
− C c − C γα ⎤
⎥
eut
− C αγ
⎦⎥
eut
)
12
(18)
The diffusivity of carbon in austenite is calculated following the work of Bhadeshia [26]. In
this model it is assumed that the average carbon content of austenite grains at each
temperature during the transformation, C , is given by the equilibrium carbon concentrations
corresponding to the (α+γ)/γ phase boundary of the equilibrium diagram of the steel. Its value
is calculated according to the procedure reported by Shiflet et al. [27]. Also, a linear variation
of the carbon content in the center of the grains with temperature is considered. C is assumed
to be related to the carbon concentration in the center of the austenite grains and at the
interface as follows,
C =
C γα + C c
2
(19)
Considering spherical austenite grains whose radius vary according to equation (16) and
taking into account Avrami theory for the relation between real and extended volume [14], the
volume fraction growth rate of the transformation can be written as,
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dVγα
= ⎡⎢6π
dt
⎣
− Vγα ⎞⎟⎤
⎥
⎟⎥
Vα o
⎠⎦
13⎡
⎛ Vα
2 6
o 2⎤ ⎢
η NV ⎥ − ln⎜ o
⎜
( )⎦
⎢
⎣
⎝
13
(Vα
o
− Vγα
)
(20)
If we consider a constant rate for the heating condition, each time interval can be expressed as
follows,
•
•
•
t = ∆T T = (T − Ac P ) T ;
dt = dT T
(21)
Introducing equation (21) into (20), separating variables and integrating, an expression for the
volume fraction of austenite transformed during heating is obtained,
α
γ
V
[
( )]
⎧
⎡ 6π 2 N o
⎪
V
= Vα o ⎨1 − exp ⎢−
• 3 2
⎢
⎪⎩
T
⎣
2 12
⎛2 T 2 ⎞
⎜ ∫Ac η dT ⎟
⎝3 P
⎠
32
⎤⎫
⎥ ⎪⎬
⎥⎪
⎦⎭
(22)
where Vγα and Vα o are the volume fraction of austenite transformed from ferrite and the initial
volume fraction of ferrite, respectively, and NVo is the number of austenite grains per unit
volume of ferrite at the end of the dissolution of pearlite which is given by,
NVo =
VPo 1
Vα o T•
T
∫Ac
1
•
(
)
N V Po − VγP dT
(23)
•
where VγP , V Po and N are the volume fraction of austenite transformed from pearlite
(equation (9)), the initial volume fraction of pearlite and the nucleation rate per unit volume of
pearlite (equation (3)), respectively.
Experimental Validation of Kinetics of Non-Isothermal Austenite Formation in Steels With a
Ferrite Plus Pearlite Initial Microstructure.
Figure 4 shows the experimental and calculated austenite formation kinetics plotted as a
function of temperature for four different heating rates in MIXT steel. In all cases,
experimental results for the austenite volume fraction are in good agreement with the
predicted values from the model and its accuracy could be considered excellent (higher than
90% in square correlation factor).
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Figure 4. Experimental and calculated kinetics results for austenite formation under four
continuous heating conditions in MIXT steel.
4. CONCLUSIONS
1. Theoretical knowledge regarding the isothermal formation of austenite from pure and
mixed initial microstructures has been used to develop a model for the non-isothermal
austenite formation in a low-carbon low-manganese steels with a mixed initial
microstructure consisting of ferrite and pearlite.
2. A mathematical model applying the Avrami's equation has been used to reproduce the
kinetics of the pearlite-to-austenite transformation during continuous heating. The model
considers two functions, f and y, which represent the influence of structure and heating
rate on the transformation kinetics.
3. A model describing the transformation process of a mixture of carbon enriched austenite
(obtained from the dissolution of pearlite) and ferrite, during continuous heating has been
developed. This process is controlled by the volume diffusion of carbon in austenite.
4. the complete model for austenitization of mixed microstructures has been validated
finding a good agreement between experimental and calculated results with an accuracy
higher than 90% in square correlation factor.
Acknowledgements
The authors acknowledge financial support from the Commission of the European
Communities (ECSC 7210-PR-349) and from the Spanish Ministerio de Ciencia y Tecnología
(MAT2002-10808 E).
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