Sha W., Malinov S. Titanium Alloys: Modelling of Microstructure, Properties and Applications
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Titanium alloys: modelling of microstructure188
where k
0
is the pre-exponential factor, Q is the activation energy and R
is the gas constant.
In general, all the adaptations of JMA have been successfully applied to
describe the kinetics of phase transformations during continuous heating,
and mostly for the processes of crystallisation of amorphous materials. This
is a relatively simple case compared with phase transformation taking place
during continuous cooling, because during heating, both nucleation and growth
rates increase with temperature.
5%
10%
25%
50%
75%
90%
95%
10 100 1000 10000
Time (sec.)
(e)
T
(°C)
1100
1050
1000
950
900
850
800
5%
10%
25%
50%
75%
90%
95%
10 100 1000 10000
Time (sec.)
(f)
T
(°C)
950
900
850
800
7.14
Continued
Johnson–Mehl–Avrami method adapted to continuous cooling 189
On cooling, the nucleation rate is determined by a Boltzmann type equation,
having an activation energy which decreases more than linearly with
temperature. This gives a rapidly increasing nucleation rate as the undercooling
increases (i.e. as temperature decreases). The growth rate, in contrast, is
controlled by an activation energy which is considered to be temperature
independent, and hence the growth rate decreases as the undercooling increases
(i.e. as temperature decreases). These opposing factors give an overall
transformation rate.
Taking these into account, it can be concluded that the methods described
for tracing the course of phase transformation during heating cannot be
directly applied to the cases of phase transformations during continuous
cooling. As an alternative approach, we replace continuous cooling T-t path
with the sum of small, consecutive isothermal steps. This concept has been
used before, to study the kinetics of different phase transformation from
DSC data. We will follow this approach to describe the kinetics of the phase
transformations during continuous cooling in titanium alloys.
The T-t path of continuous cooling is replaced by the sum of consecutive
isothermal steps (see Fig. 7.15), at temperatures T
1
, T
2
, T
i
(note that cooling
rate is a constant). For each isothermal step, JMA theory for isothermal
phase transformations, Eq. [6.1], is applied. Hence, for cooling from point 0
to point 1, the following can be written:
fkt
n
11
= 1 – exp(– )
1
∆
[7.4a]
where f
1
is degree of transformation from 0 to 1 (see Fig. 7.15), k
1
and n
1
are
JMA kinetics parameters at temperature T
1
and ∆t is isothermal time.
For tracing the transformation from 0–2, the principle of additivity must
be adhered to. Thereby, the first fictitious time for point 1 is calculated using
t
f
k
n
2
*
1
2
=
ln(1 – )
–
2
[7.4b]
where
t
2
*
refers to the time after which the amount f
1
would have been
transformed if the whole transformation were at an isothermal temperature
T
2
with kinetics parameters k
2
and n
2
. Thereafter, the time is increased by the
increment of ∆t and for interval 0–2, one may write:
fktt
n
222
*
= 1 – exp(– ( + ) )
2
⋅∆
[7.4c]
From the above reasoning, the general case for interval 0–i is as follows:
fktt
iii
n
i
= 1 – exp(– ( + ) )
*
⋅∆
[7.4d]
where
t
f
k
i
i
i
n
i
*
–1
=
ln(1 – )
–
[7.4e]
Titanium alloys: modelling of microstructure190
0
T
1
(
k
1
,
n
1
)
1
2
i
T
2
(
k
2
,
n
2
)
T
i
(
k
i
,
n
i
)
0 100 200 300 400
Time (sec.)
f
1
f
2
0.2 0.4 0.6 0.8
f
i
Transformed fraction
T
(°C)
960
940
920
900
880
860
960
940
920
900
880
860
7.15
Schematic diagram for the approximation of continuous cooling as the sum of short time isothermal holding.
∆
t
Johnson–Mehl–Avrami method adapted to continuous cooling 191
These calculations present a numerical solution of the general expression of
the concept of additivity.
On the basis of the above description, a computer program can be created
to derive the optimised JMA kinetic parameters at different temperatures
from the DSC experimental data (see Fig. 7.16). The input data are the
degree of transformation as a function of the temperature for cooling rates of
10, 20, 30 and 40 °C/min, etc. The program contains loops for the kinetic
parameters at the different temperature stages (k
1
, k
2
, …, k
i
and n
1
, n
2
, …, n
i
).
The experimental degree of transformation is compared with the calculated
degree of transformation and errors are minimised using a sum-squared error
technique.
7.16
Flow diagram of program for calculation of JMA kinetic
parameters.
Input data
Transformed fraction as a
function of the temperature for
different cooling rates,
f
=
f
(
T
)
k
1
n
1
k
i
n
i
n
1
=
n
1
+∆
n
k
1
=
k
1
+∆
k
k
i
=
k
i
+∆
k
n
i
=
n
i
+∆
n
t
f
k
i
i
i
n
i
*
–1
=
ln(1 – )
–
fktt
iii
n
i
= 1 – exp (– ( + ) )
*
⋅∆
min ∑ (
f
exp
–
f
calc
)
2
Output
k
=
k
(
T
)
n
=
n
(
T
)
Titanium alloys: modelling of microstructure192
For the calculations, in these particular cases, (the β to α + β transformation
in Ti-6Al-4V, Ti-6Al-2Sn-4Zr-2Mo-0.08Si, Ti-8Al-1Mo-1V, IMI 834 and
IMI 367 and the γ phase formation in Ti-46Al-2Mn-2Nb), it is accepted that
n does not vary with the temperature (n
1
= n
2
= n
i
). The value of n depends
largely on the growth geometry, and therefore it should change only when
this geometry varies. The output of the program to derive JMA kinetic
parameters is n = 1.13 for Ti-6Al-4V, n = 1.01 for Ti-6Al-2Sn-4Zr-2Mo-
0.08Si, n = 0.82 for Ti-8Al-1Mo-1V, n = 0.95 for IMI 834, n = 1.15 for IMI
367 and n = 2.2 for Ti-46Al-2Mn-2Nb, and k = k(T) (see Fig. 7.17).
These n values imply that the preferable nucleation sites are β grain
boundaries in the cases of β to α + β transformation in Ti-6Al-4V, Ti-6Al-
Reaction rate constant,
k
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
840 860 880 900 920 940 960 980
T
(°C)
(a) Ti-6Al-4V
Reaction rate constant,
k
0.06
0.05
0.04
0.03
0.02
0.01
0
840 860 880 900 920 940 960 980 1000
T
(°C)
(b) Ti-6Al-2Sn-4Zr-2Mo-0.08Si
7.17
Calculated rate constant (
k
) as a function of the temperature. (a)
to (e) β to α + β or β to α phase transformation; (f) γ phase formation
in Ti-46Al-2Mn-2Nb.
Johnson–Mehl–Avrami method adapted to continuous cooling 193
Reaction rate constant,
k
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
850 900 950 1000 1050 1100
T
(°C)
(c) Ti-8Al-1Mo-1V
Reaction rate constant,
k
0.2
0.15
0.1
0.05
0
850 900 950 1000 1050 1100
T
(°C)
(d) IMI 834
Reaction rate constant,
k
0.06
0.05
0.04
0.03
0.02
0.01
0
850 900 950 1000
T
(°C)
(e) IMI 367
7.17
Continued
Titanium alloys: modelling of microstructure194
2Sn-4Zr-2Mo-0.08Si, Ti-8Al-1Mo-1V, IMI 834 and IMI 367. This is
characteristic of the phase transformation during cooling in titanium alloys,
and is confirmed by metallographic examinations in Section 7.4, as well as
metallographic examinations at different stages of the β to α + β transformation.
Similar n values have been reported for titanium alloys 6-2-4-6, β-CEZ and
10-2-3 (Bein and Bechet, 1996). Bein and Bechet (1996) found that for 6-2-
4-6, n = 1 for the temperature range from β-transus (940 °C) to 240 °C below
β-transus, and for β-CEZ, n was in the range of 1–1.2 for temperature range
from β-transus (890 °C) to 290 °C below β-transus.
The value of k for the transformations in the alloys varies vastly with
temperature (see Fig. 7.17). In the case of β to α + β transformation in Ti-
6Al-4V, Ti-6Al-2Sn-4Zr-2Mo-0.08Si, Ti-8Al-1Mo-1V, IMI 834 and IMI 367,
it increases significantly when the temperature decreases from β-transus to
850 °C. In the case of γ phase formation in Ti-46Al-2Mn-2Nb, it increases
significantly when the temperature decreases from 1300 to 1270 °C, and
then decreases when the temperature is below 1270 °C. A similar tendency
of k(T) was reported for austenite to pearlite phase transformation in steels
(Homberg, 1996). The way of variation of k means that, in the above
temperature range, the nucleation rate controls the overall transformation
rate. From the DSC data using a high cooling rate (50 °C/min), there is some
evidence of a decrease of k when the temperature decreases to below 850 °C.
These observations are in good agreement with C-curves of the experimental
TTT diagrams for Ti-6Al-4V (Chapter 14) and Ti-6Al-2Sn-4Zr-2Mo-0.08Si
(Rosenberg et al., 1991), and the TTT diagram predicted from an artificial
neural network model for the Ti-6Al-2Sn-4Zr-2Mo-0.08Si alloy (Chapter 14).
As the next step, we calculate the degree of transformation for different
cooling rates, using the derived values of n and k. Good agreement between
Reaction rate constant,
k
3·10
–5
2.5·10
–5
2·10
–5
1.5·10
–5
1·10
–5
5·10
–6
1240 1250 1260 1270 1280 1290 1300
T
(°C)
(f) Ti-46Al-2Mn-2Nb
7.17
Continued
Johnson–Mehl–Avrami method adapted to continuous cooling 195
Experimental
Calculated
10 °C/min
20 °C/min
30 °C/min
40 °C/min
840 860 880 900 920 940 960 980
T
(°C)
(a) Ti-6Al-4V
Degree of transformation
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Experimental
Calculated
10 °C/min
20 °C/min
40 °C/min
50 °C/min
840 860 880 900 920 940 960 980 1000
T
(°C)
(b) Ti-6Al-2Sn-4Zr-2Mo-0.08Si
Degree of transformation
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
30 °C/min
7.18
Experimental and calculated degree of transformations for
different cooling rates. (a) to (e) β to α + β or β to α phase
transformation; (f) γ phase formation in Ti-46Al-2Mn-2Nb.
calculated and experimental degree of transformation is shown (Figs. 7.18
and 7.19). In the case of Ti-6Al-4V, there is some deviation, at approximately
0.05 level, for temperatures above 900 °C and cooling rates above 20 °C/
min. The main reason for the deviation at high temperatures is because, as
stated before, the transformation start is not clearly defined in the DSC
curves and a start temperature of 970 °C is attributed, for all cooling rates.
Also, the experimental error is larger at higher cooling rates. Nevertheless,
agreement between experimental and model results is more than acceptable.
Titanium alloys: modelling of microstructure196
Experimental
Calculated
Degree of transformation
1
0.8
0.6
0.4
0.2
0
850 900 950 1000 1050 1100
T
(°C)
(c) Ti-8Al-1Mo-1V
Experimental
Calculated
Degree of transformation
1
0.8
0.6
0.4
0.2
0
850 900 950 1000 1050 1100
T
(°C)
(d) IMI 834
Experimental
Calculated
Degree of transformation
1
0.8
0.6
0.4
0.2
0
850 900 950 1000
T
(°C)
(e) IMI 367
7.18
Continued
Johnson–Mehl–Avrami method adapted to continuous cooling 197
These parameters can be used to trace the course of the phase transformations
in titanium alloys in real practice for any, not necessarily constant, cooling
path.
7.7 Simulation and monitoring of transformations
on continuous cooling
The JMA kinetic parameters are derived from DSC or DTA data with constant
cooling rates. However, once obtained, these parameters can be used to trace
the course of β to α transformation and γ phase formation in real practice for
a non-uniform cooling rate. Based on the derived kinetic parameters, we
have developed models for monitoring the β to α transformation and the γ
phase formation during continuous cooling with an arbitrary cooling path
(Fig. 7.20). The input for the model is the cooling path, which can be user-
defined or input from a real furnace. The kinetic parameters as well as the
data on the thermodynamic equilibria are incorporated in the model. Utilising
the cooling laws, kinetic parameters and thermodynamic equilibria, the
transformation process is calculated and displayed. The calculation procedure
is similar to the calculations previously described. The arbitrary cooling path
is digitised and presented as a sum of consecutive isothermal steps (note that
since the cooling rate is not constant the intervals are not equal with respect
to time). The calculations are then performed using the JMA models, taking
into account the principle of additivity and the phase equilibrium conditions.
In this way, the transformation process can be traced for any cooling path.
One example generated from the model is demonstrated in Fig. 7.21. The
model may not work with sufficient accuracy for high cooling rates (>100
°C/min), where martensitic transformation may be involved.
Experimental
Calculated
Degree of transformation
1
0.8
0.6
0.4
0.2
0
1220 1240 1260 1280 1300
T
(°C)
(f) Ti-46Al-2Mn-2Nb
15 °C/min
7.18
Continued