Sha W., Malinov S. Titanium Alloys: Modelling of Microstructure, Properties and Applications
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Neural network models and applications in property studies 401
Microstructure
The morphology of the alloy is governed by its TMP conditions. In this
model, 12 different microstructure types are described. Since the microstructure
of an alloy is a qualitative measure, and since neural networks operate with
numeric entries only, the microstructure types need to be represented
numerically. This is achieved by assigning each microstructure a reference
number, which will be used for the model input (Table 15.6).
A considerable amount of the available data does not describe the
microstructure (about 40% of total data entries). However the remaining
data (about 60% of total data entries) cover a wide variety of attainable
microstructures, and are useful. The overlap in microstructure types is due to
the different terminology used in different sources. For example, ‘fully
transformed’ microstructure is similar to ‘fully lamellar’. These different
terminologies are preserved in modelling however, to avoid possible distortion
from original literature data. It should be noted that the difference in the
numerical value, e.g. 5 for ‘fully transformed’ and 9 for ‘fully lamellar’ is
not proportional to the difference in the nature of the corresponding
microstructures.
Temperature
A wide range of temperature entries is used in the model, ranging from 15 to
1200 °C. Room temperature is assumed to be 25 °C. In the cases where the
tensile test temperature is unknown, it is assumed that a room temperature
(RT) tensile test is used. Most (about three quarters of total data entered)
data range from 500 to 1200 °C, so the model will work with sufficient
accuracy within these limits. It must also be noted that to form the sub-
Table 15.6
Microstructures used in the titanium aluminides neural
network and their numerical equivalents
Alloy microstructure Reference number
Unknown 0
Duplex 1
Single-phase 2
Equiaxed bimodal 3
Fully transformed 5
Fine lamellar grains 6
Coarse lamellar grains 7
Fully lamellar 9
Nearly lamellar 10
Primary gamma grains: fine grains 11
Primary gamma grains: coarse grains 12
Equiaxed gamma grains 13
Titanium alloys: modelling of microstructure402
models to predict the tensile properties, not all the temperature data entries
were used, and only those that were relevant to the particular output were
included. This means whilst overall the model functions with sufficient accuracy
for a range of 500–1200 °C, for each individual output the working temperature
range differs from the overall working range (Table 15.5).
Output parameters
The tensile properties predicted by the model are elastic modulus, ultimate
stress, elongation and reduction in area. There are four models, one for each
tensile property. However, in the text, sometimes the word ‘model’ is used to
refer to the four models collectively.
Training of the artificial neural network
An important factor in the successful training of a network is the optimisation
technique. Recent advances have led to much improved methods of numerical
optimisation. There are non-linear relationships in alloys between their
processing parameters and their mechanical properties.
Before the network was trained, the input and output variables were
normalized within the range –1 to 1. For the various models, the number of
neurons in the hidden layer varied from 2 to 9 neurons. A Bayesian
regularisation technique was used to improve the generalisation of the models.
Performance of the neural network
The performance of the neural network was checked after training (Fig.
15.19). The analysis of the network response is plotted in the form of linear
regression analysis between the network output (predictions) and the
corresponding targets (experimental data). The neural network predicts the
values to a high degree of accuracy. Similar graphs are obtained for other
tensile properties.
The correlation coefficient (R) values for the test data sets only, for all
outputs, are shown in Table 15.7. Five model training runs were made for
each output, with the same neural network architecture but different division
of training and testing data sets. The model with the highest R value (as
given in Table 15.7) was chosen as the final model.
Further analysis of the performance accuracy was carried out using statistical
analysis of the error of neural network predictions. Neural network predictions
were compared to the corresponding experimental values. The relative errors
were then calculated, using a relationship similar to Eq. [15.1].
The results are displayed graphically, in a number versus error plot, using,
as an example, the elastic modulus (Fig. 15.20). For most of the data set, the
Neural network models and applications in property studies 403
Data points
A = T
Best linear fit
Data points
A = T
Best linear fit
Data points
A = T
Best linear fit
R = 0.989
Train
80 100 120 140 160 180
Experimental elastic modulus (T)
Neural network predictions (A)
180
170
160
150
140
130
120
110
100
90
Test
R = 0.997
110 120 130 140 150 160 170
Experimental elastic modulus (T)
Neural network predictions (A)
170
160
150
140
130
120
110
R = 0.99
80 100 120 140 160 180
Experimental elastic modulus (T)
Neural network predictions (A)
180
170
160
150
140
130
120
110
100
90
All
15.19
Performance of the
neural network for simulation
of elastic modulus of titanium
aluminides, plotting the neural
network versus experimental
values for the training, test,
and whole data sets.
T
is the
experimental elastic modulus,
in GPa.
A
is the predicted
elastic modulus from the NN,
in GPa.
Titanium alloys: modelling of microstructure404
error of prediction is within ±2%, which is a high degree of accuracy, of this
particular model.
The other models exhibit reasonable degrees of error. The models give an
overall satisfactory performance, and their output predictions are of an
acceptable accuracy.
15.3.2 Comparison of the prediction and the
experimental data
Using trained artificial neural networks, a model for the prediction of the
tensile properties of titanium aluminides was created. It can be used to
–8 –6 –4 –2 0 2 4 6 8
Relative error (%)
Number
45
40
35
30
25
20
15
10
5
0
15.20
Statistical analysis of the error of the neural network
predictions for elastic modulus of titanium aluminides.
Table 15.7
Regression coefficients for each titanium aluminide tensile
property model
Property Test accuracy (
R
) Number of neurons
in hidden layer
Elastic modulus 0.997 2
Ultimate strength 0.957 9
Elongation 0.943 7
Reduction in area 0.971 3
Neural network models and applications in property studies 405
predict, with sufficient accuracy, each of the four properties within the ranges
given in Table 15.5.
In the following, tensile properties for different gamma titanium aluminides,
with different microstructures and test temperatures, are predicted by the
model and then analysed. All the results are compared with experimental
data from the relevant literature, with positive outcomes.
Room temperature properties
Room temperature tensile strength is selected for comparison purposes (Fig.
15.21). The accuracy of the other models is reasonably similar.
There is good agreement between the predicted values from the model
and the experimental data for most of the outputs. Note that the reduction in
area could not be compared in this case since the database for this property
contained no data from room temperature tests. Some poor correlation is due
to a lack of data, causing insufficient NN accuracy. This accuracy cannot be
improved through training if no data are available.
Properties at elevated temperatures
It is also required that the model predict, to within a sufficient degree of
accuracy, the tensile properties at elevated temperatures, e.g. 200–1200 °C.
This is important for the use of the alloys in aerospace applications. The
tensile properties of various alloys are calculated across temperature ranges
from 20 to 1200 °C, and compared with experimental data available for
these temperatures in Figs. 15.22–15.24.
Model prediction
Experimental
Ti-54Al single
phase
Ti-47Al-1Cr-
1V-2.6Nb
fully lamellar
Ti-48Al
equiaxed
bimodal
Ti-47Al-1Cr-
1V-2.5Nb
duplex
Ultimate strength (MPa)
1400
1200
1000
800
600
400
200
0
15.21
Accuracy of the model prediction for ultimate strength of
titanium aluminides, at room temperature, when compared with
experimental data.
Titanium alloys: modelling of microstructure406
Model prediction
Experimental
Ultimate strength (MPa)
350
300
250
200
150
100
50
0
1000 °C 1100 °C 1200 °C
15.22
Comparison between predicted and experimental tensile
strength for Ti-46.5Al-5Nb alloy with a nearly lamellar microstructure,
at different temperatures.
Model prediction
Experimental
Reduction in area (%)
70
60
50
40
30
20
10
0
1000 °C 1100 °C 1200 °C
15.23
Comparison between predicted and experimental reduction in
area for Ti-47Al-2Cr alloy with a nearly lamellar microstructure, at
different temperatures.
Model prediction
Experimental
0 200 400 600 800 1000 1200
Temperature (°C)
Elastic modulus (GPa)
180
170
160
150
140
130
120
110
100
90
15.24
Correlation between predicted and experimental elastic
modulus for Ti-40Al with increasing temperatures (20–1000 °C).
Neural network models and applications in property studies 407
As can be seen, there is a good correlation between the predicted and the
experimental values. This was also found with most of the other tests. It is
useful to note that the strength properties (ultimate strength and elastic modulus)
obtained from the model reduce as the temperature increases. The plasticity
properties (elongation and reduction in area) obtained from the model increase
as the temperature increases. Therefore the effect of the temperature on the
model prediction of the tensile properties is consistent with what is expected
from a metallurgical point of view. It is confirmed further by the experimental
data. The output of the model allows us to obtain the safe working ranges of
the alloys. For example, we can see from the predicted elastic modulus for
Ti-40Al (Fig. 15.24) that no serious reduction in this parameter is apparent
from 0 to 600 °C, after which it reduces rapidly. This is confirmed by the
curve obtained from experimental data. Therefore we can say, in this instance,
that the alloy has a safe working range of 0–600 °C for structural applications
requiring a high elastic modulus. This is vitally important for the use of such
an alloy in an aircraft.
Overall, the model performs well at elevated temperatures, giving sufficiently
accurate predictions.
Effect of the microstructure conditions
The microstructure of the alloy is of considerable importance to its mechanical
properties. The type of heat treatment can determine the microstructure of
the alloy, and hence its ductility or strength. In order to understand the effect
of microstructure types, the model is used for three alloys to predict the
ultimate strength achieved with different microstructures (Fig. 15.25).
15.25
Correlation between predicted and experimental ultimate
strength for three titanium aluminide alloys with various
microstructures.
Model prediction
Experimental
Ti-48Al-2Cr- Ti-48Al-2Cr- Ti-48Al-1V- Ti-48Al-1V- Ti-48Al-2Nb- Ti-48Al-2Nb-
2Nb duplex 2Nb fully 0.2C duplex 0.2C equiaxed 2Mn equiaxed 2Mn fully
750 °C transformed 20 °C bimodal bimodal lamellar
750 °C 20 °C 650 °C 650 °C
Ultimate strength (MPa)
600
500
400
300
200
100
0
Titanium alloys: modelling of microstructure408
The model is in good agreement with the experimental data. However
microstructures are described qualitatively, and have no quantitative
information. The particular thermomechanical processing (TMP) conditions
which produce the various microstructures, e.g. heating temperature, time,
heating and cooling rates and environments are not taken into account in the
model. Thus the model assumes that all alloys have received the optimal
TMP conditions required to obtain the corresponding microstructures. The
model would be more accurate, and of greater use to metallurgists, if it
included the various degrees of TMP as well as the resulting microstructure.
It would allow the user to estimate the TMP conditions needed to obtain the
desired microstructure, and hence the desired properties. However, there are
insufficient data sources available in the literature to do this.
Effect of alloy composition
The composition of the alloy has a major influence on its mechanical properties.
Compositions for a wide range of alloys were used in the model. Attempts to
simulate the correlation between the proportion of alloying elements and the
alloy tensile properties showed that the experimental data were sufficiently
accurate to allow the model to correctly simulate the effect of these alloying
elements on the properties. The influence of oxygen was not considered in
this model. This was assumed to be a constant value throughout for
developmental purposes. However, the oxygen content in alloys may be
desirable for certain applications, and would give a more accurate result if
included. It would allow the user to obtain the optimum oxygen content
within the alloy for certain properties, and enable them to decide the appropriate
heat treatment procedure necessary to achieve it.
15.3.3 Graphical user interface
A graphical user interface (GUI) such as the ones shown in Chapter 13 and
Section 15.1.7 was created for easy user interface with the model. When
alloy composition, test temperature range and microstructure are entered, the
model will plot a graph depicting the specified property variance over the
temperature range. This allows the user to easily see the temperature dependence
of any of the four tensile properties involved in the model for a selection of
microstructures and any alloy composition. This allows various combinations
of the inputs to be examined.
15.3.4 Summary
An artificial neural network has been created for the prediction of the tensile
properties of gamma-based titanium aluminides using existing experimental
Neural network models and applications in property studies 409
data from available literature. The model has been used to study the influence
of the three most important input conditions, viz. alloy composition,
microstructure and temperature, on four tensile properties. These properties
are ultimate stress, elastic modulus, elongation, and reduction in area. Further
input of strain rate is used in the modelling of reduction in area. Most of the
models created are of acceptable accuracy within the experimental data range.
A GUI has been created for use with the model. The program and the model
can be used for:
• prediction of tensile properties for gamma titanium aluminide alloys as
functions of their chemical composition, microstructure and working
temperature;
• investigation of the interaction of these factors on the properties;
• reducing the amount of sample testing needed to obtain these properties.
Although the model produced for the tensile properties of gamma-based
titanium aluminides has met with some success, there are several future
additions that could be included to further improve the accuracy of prediction.
Most importantly, new inputs could be added to the model. The actual TMP
conditions used to achieve the described microstructures has been omitted
from this model, and would provide benefit for users wishing to find the
optimum TMP inputs to achieve their desired properties. Also the oxygen
content within the alloys, an aspect affected by TMP conditions, could be
included. More up-to-date experimental data could be included which would
improve the less successful models. By increasing the amount of input data,
there should be an improved training accuracy.
Alternative work relating to this topic would be the formation of corrosion
models for similar alloys. The benefit of gamma-based titanium aluminides
is their improved oxidation resistance, so it would be of great benefit to
researchers and engineers if such a model was attempted. Then all available
data on gamma titanium aluminides, including mechanical properties and
corrosion resistance, could be collected on one master database and simulated
on a master model. This may lead to an improved use of these materials
within industry by reducing the cost of research and development into them,
offering them as a viable alternative to conventional superalloys.
15.4 Reference
Winstone M R (ed) (2001), Titanium Alloys at Elevated Temperature: Structural Development
and Service Behaviour, London; IoM Communications.
Titanium alloys: modelling of microstructure410
15.5 Appendix
A list of the alloys used for training and test of artificial neural network for
prediction of mechanical properties of titanium alloys:
Ti-6Al-2Sn-1.5Zr-1Mo-0.35Bi-0.1Si, Ti-6Al-2Nb-1Ta-0.8Mo, Ti-6Al-2Nb-
1Ta-1Mo, Ti-6Al-1Mo-2Nb-1Ta, Ti-3Al-2.5V, Ti-5Al-2.5Sn, Ti-5Al-2Zr-2Mo-
0.25Si, Ti-6Al-2Sn-4Zr-2Mo, Ti-6Al-2Sn-4Zr-2Mo-0.1Si, Ti-1Sn-5Zr-2.25Al-
1Mo-0.25Si, Ti-8Al-1Mo-1V, IMI 834, IMI 417, Titanium 1100, Ti-10V-
2Fe-3Al, Ti-11.5Mo-Zr-4.5Sn, Ti-3Al-8V-6Cr-4Mo-4Zr, Ti-12V-2.5Al-2Sn-
6Zr, Ti-15Mo-5Zr, β-21S, Ti-11.5V-2Al-2Sn-11Zr, Ti-13V-11Cr-3Al, Ti-13V-
2.7Al-7Sn-2Zr, Ti-15V-3Cr-3Al-3Sn, Ti-15Mo-5Zr-3Al, Ti-8Mo-8V-2Fe-3Al,
β-CEZ, Ti-4.5Al-5Mo-1.5Cr (Corona 5), Ti-8Mn, Ti-5Al-2Sn-2Zr-4Mo-4Cr
(Ti-17), Ti-6Al-2Sn-2Zr-2Mo-2Cr-0.25Si, Ti-4Al-4Mo-2Sn-0.5Si (IMI 550),
Ti-6Al-2Sn-4Zr-6Mo, Ti-6Al-4V, Ti-6Al-6V-2Sn, Ti-7Al-4Mo, Ti-5Al-5Sn-
2Zr-2Mo-0.25Si, Ti-5Al-3.5Sn-3Zr-1Nb-0.3Si, Ti-5Al-6Sn-2Zr-1Mo-0.25Si,
Ti-6Al-1.7Fe-0.1Si, Ti-4.5Al-3V-2Mo-2Fe, Ti-6Al-7Nb, Ti-4Al-4Mo-4Sn-
0.5Si, Ti-4Al-3Mo-1V, Ti-5Al-1.5Fe-1.4Cr-1.2Mo, Ti-5Al-2.5Fe, Ti-6.4Al-
1.2Fe, Ti-2Fe-2Cr-2Mo, Ti-8V-5Fe-1Al, Ti-16V-2.5Al, Ti-3Al-1.5Cr-1.5Fe,
Ti-6Al-2Fe, Ti-6Al-2Fe-0.1Si, Ti-6Al-1Fe-1Cr, Ti-8Al-2Fe, VT14, VT15,
VT16, VT18, VT20, VT22, OT4, VT1-00, VT3-1, VT5, VT5-1, VT-6, VT-
6C, VT-8, VT-9.