A Neural Network Model of the Creep Strength of Austenitic Stainless Steels
T. Sourmail, H. K. D. H. Bhadeshia and D. J. C. MacKay*
Department of Materials Science and Metallurgy, University of Cambridge
Pembroke Street, Cambridge CB2 3QZ, U.K.
*Department of Physics
Cavendish Laboratory, University of Cambridge
Mat. Sci. Tech. 2002:18-6 p655-663
The creep rupture life and rupture strength of austenitic stainless
steels have been expressed as functions of chemical composition, test
conditions, stabilisation ratio and solution treatment
temperature. The method involved a neural network analysis of a vast
and general database assembled from published data.
The outputs of the model have been
assessed against known metallurgical trends and other empirical
modelling approaches. The models created are shown to capture
important trends and to extrapolate better than conventional
Austenitic stainless steels are commonly used in the power generation
industry at temperatures greater than 650 C and stresses of 50
MPa or more, and are expected to remain in service for more than
100,000 h. Such time periods are seldom accessible experimentally and
long-term properties are often extrapolated from shorter term
tests conducted at high stresses. Therefore great care is needed in
extrapolating the experimental data during design.
A further difficulty in trying to predict the long term properties of
austenitic steels is the strong influence of alloying elements and
their numerous interactions. This explains why most of the empirical
approaches are restricted to limited ranges of compositions, for
example, equation 1 has been proposed for the
104 h creep rupture stress (
) of a AISI 316 at 650 C,
while equation 2 is for a 304 steel at
the same temperature :
where the concentrations are in wt%.
The range for each variable is not given in ,
but it appears that equation 2 does not even cover
the range of composition that separates an AISI 304 steel from an AISI 316.
Such equations often come from limited studies and therefore
only address the role of a few alloying elements.
As a consequence of the restricted amount of data on which
they are based, few account for interactions between variables.
Finally, they do not account explicitly for
the effects of temperature, time and stress.
Neural networks represent a more general regression method,
which ameliorates most of the problems encountered with linear
regression. In this study, neural network analysis was applied to a
database covering a vast range of compositions of austenitic stainless
steels to estimate the creep rupture life and the creep rupture
stress as a function of many parameters.
A neural network is a parameterised non-linear model which can be used
to perform regression, in which case, a very flexible, non-linear
function is fitted to experimental data. The details of this method
have been reviewed elsewhere [2,3], but
it is nevertheless useful to introduce its main features.
A three layer feedforward network similar to
the one used in this work. The activation function of the neurons in
the second layer is a
tanh, and is linear in the second one. The complexity
of the model is controlled by the number of neurons in the second
layer, also called hidden units.
Simple three layer feedforward networks are used, as the one shown in
figure 1. The activation function for the neurons in
the second layer (equation 3) is tanh while it is
linear (equation 4) in the third one.
xi are the inputs and wi the parameters, or weights, which
define the network. The biases are treated internally as
weights associated with a constant input set to unity. Any
nonlinear function can be used at the hidden units (as long as it is
continuous and differentiable), and tanh is a standard choice for such
networks. A linear function for the output is a simple choice to
ensure that all values can be taken (tanh would limit the output to
, for example).
The complexity of such models scales with the number of neurons in the
second layer, most often referred to as the hidden units. The neural
network can capture interactions between the inputs because of the
non-linearity of the activation function. The nature of these interactions is
implicit in the weights but they are often difficult to interpret
directly. The best way to identify the interactions is to use the network to make
predictions and see how these depend on various combinations of input.
Many training methods involve finding the weights which minimise an
objective function, typically:
where ED is the overall error, and EW the regulariser, used
to force the network to use small weights (equations
5). and are control parameters which
largely influence the complexity of the model. t(i) is the target
for the set of inputs , while y(i) is the
corresponding network output.
The method used in this study, developed by MacKay , is
based on Bayesian probability theory and treats learning as an
Rather than trying to identify the best set of weights,
the algorithm infers a probability distribution for the weights
from the data presented. When making predictions, the
variety of solutions corresponding to different possible sets of
weights are averaged using the probabilities of these sets of
weights, a process called marginalising.
A major consequence is that it is possible to
quantify the uncertainty of fitting: if the inferred distribution is
sharply peaked in the weight space, the most probable set will
give by far the largest contribution to the prediction and
alternative solutions will have little importance. As a consequence,
the prediction will be associated with a small uncertainty.
If on the contrary, the data are such that different sets of weights
are similarly probable, alternatives will contribute in
similar proportions and the error bar will be large, as typically
occurs in regions of the input space where data are scarce or
In this context, the performances of different models are best
evaluated using the log predictive error (LPE) as defined below.
This error penalises wild predictions to a lesser extent when they
are accompanied by appropriately large error bars.
is related to the uncertainty of fitting for
the set of inputs .
Because of the great flexibility of the functions used in the network,
there is a possibility of overfitting data.
Two solutions are implemented which contribute to avoid overfitting.
The first is contained in the algorithm due to MacKay: the complexity
parameters and are inferred from the data, therefore
allowing automatic control of the model complexity.
The second resides in the training method. The database is equally divided
into a training set and a testing set. To build a model, about 150
networks are trained with different numbers of hidden units and seeds,
using the training set; they are then used to make predictions on the
unseen testing set and are ranked by LPE.
illustrates the behaviour of the error on the training and the testing
set. Because it is possible to obtain a near perfect fitting, the
error on the training set is always decreasing with increasing
complexity. The error on the testing set decreases at first, as the
fitting improves, but increases again when overfitting occurs.
(A) When a model has overfitted the training
data (), the error on the test data ( x ) is larger
then for an optimum model which fits the trend but not the
noise. (B) illustrates the behaviour of the error on the training
and testing sets as a function of the complexity of the model.
To ensure a good distribution of the data in the two subsets, the
database is initially randomised. The input and outputs are then
rescaled into the interval , this step is not obligatory
but is a convenient way of comparing the effect of different variables
on the output.
As is evident from the above discussion, networks with different
numbers of hidden units will give different predictions. But predictions
will also depend on the initial guess made for the probability
distribution of the weights (the prior).
Optimum predictions are often made using more than one model, by
building a committee. The prediction, , of a committee of
networks is the average prediction of its members, and the associated
error bar is calculated according to:
where L is the number of networks in the committee. Note that we now
consider the predictions for a given, single set of inputs and that the
exponent (l) refers to the model used to produce the corresponding
In practice, an increasing number of networks are included in a
committee and their performances are compared on the testing set. Most
often, the error is minimum when the committee contains more than one
The selected models are then retrained on the full database.
A large database was compiled for the creep properties of various grades of austenitic
stainless steels: AISI 304 (basic 18Cr-12Ni), AISI 316 (304 + Mo),
AISI 321 (304 + Ti), AISI 347 (304 + Nb)
and many variants designed for heat-resistant applications (for example, 316 + Ti,
Esshete 1250, etc.). It contains a total of about 3500 entries which,
as explained above, are equally distributed between the training and
Figure 3 gives an idea of the distribution of each input
against the logarithm of the rupture life.
The distribution of the different inputs
against the log of creep rupture life. This way of representing
the data should not hide the possibility of numerous non
The dataset included all of the following NRIM datasheets : 5B, 6B, 45, 28A,
16B, 26B, 14B, and most of the data published by the British
Steelmakers Creep Committee  for 304, 316, 321 and 347.
Only limited data could be extracted from publications [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21].
This is essentially because of non standard pre-test mechanical
treatments performed in order to accelerate the evolution of the microstructure.
The data set includes the following variables: test conditions
(stress and temperature), chemical composition, solution treatment temperature and
time (the latter being available in a very limited number of cases), nature of the
quench following, grain size, and logarithm of rupture life. The
minimum and maximum values are given in table
1. Because the algorithm includes an automatic
relevance detection , variables which are either redundant or found to
be irrelevant are affected a zero weight. There is therefore little to
gain in reducing the number of input variables by any other process.
The different inputs in the data set.
|Test Stress (MPa)
|Test Temperature (oC)
|log (rupture life / h)
|Solution treatment temperature (oC)
Compositional data were often missing. In such circumstances,
elements usually known to be deliberate additions were set to
zero while impurities were set to the
average of the available data (e.g. phosphorus and sulphur). There is
undoubtedly a regrettable loss of information when the amounts of
elements such as Mo or Nb present as impurities are not given, as
there is evidence that these elements have an influence .
It is usual to attempt to predict the rupture strength
for a given life time. However, the creep stress is only present as a
finite number of discrete values while the rupture life is much more
continuously spread, and therefore seemed a more appropriate target.
When training a model, the choice of input variables is of great
importance. Also, when a combination of these variables is
believed to be of particular importance, the model can be
improved by adding the combination as an explicit variable. The model
was trained on the logarithm of the rupture life rather than the
rupture life itself, and the calculated stabilisation ratio was
used, as given below:
where the concentrations are in weight %. This is because
precipitation of MX (where M is either Nb or Ti and X either C or N)
is believed to be of particular importance to the creep behaviour of
austenitic stainless steels, as will be discussed in more details
later. To avoid biasing the model, the individual variables making up
the stabilisation ratio are also included, so that a direct influence
of any of them can also be detected.
Other input variables are as given in .
The perceived level of noise
(a), the test error (b), the log predictive error (c)
of the models with increasing numbers of hidden units, the combined
test error (d) for an increasing number of models in committee, and
the performances of the best single model on seen (training set,
(e)) and unseen data (testing set (f)). (e) and (f) are plots of the
predicted rupture life (R.L.) against the experimental values, in
this case both are normalised.
Because some inputs were not always given (the solution treatment
temperature for example), about 1000 amongst the 3500 entries of the
database could not be used.
About 130 networks were trained with up to 22 hidden units and 6
different seeds. As expected, the perceived level of noise during
training decreases as the model becomes more complex.
The results of the training are shown in figure 4.
The purpose and method for building a committee model has been
discussed earlier. In this case, the optimum committee was found to have
4 members. The perceived significances for these four
models are shown in figure 5. They represent
the extent to which a particular input explains the variation of the
output, rather like a partial correlation coefficient in a multiple
linear regression analysis.
The perceived significances for
the first four networks, constituting the committee model for the
creep rupture life. S.t. stands for solution treatment.
The predictions of the final committee model (figure
6) contain a very few outliers
considering that there is a total of about 2000 points. The
improvement is clear compared with the best model alone.
The performances of the final committee model
on the whole database, for the rupture life model (a) and the creep
strength model (b).
The creep strength model was built essentially to facilitate
quantitative comparisons with literature. In this case, the target was
the stress and the life time was an input. The solution
treatment temperature was not included to allow use of the entire
The best model predictions on the training
set (a) and test set (b) for the creep rupture model.
It seems of little interest to reproduce here all the results
from the training such as test error or LPE as a function of number of hidden
units as their evolution was similar to that observed for
the creep rupture life model. In this case, the optimum number of
models in committee was found to be 12. The performance of the best
model is shown in figure 7 and that of the
committee in figure 6.
The difference between AISI grades 304 and 316 resides essentially in the
addition of about 2 wt% of molybdenum. The chromium and nickel
concentrations are smaller and larger, respectively, for AISI 316
compared to AISI 304.
Rupture stresses for 104 h at 650 C are
respectively around 80 MPa and 110 MPa .
(A) The predicted effect of Mo on (a) the
104 h and (c) the 105 h rupture stress at 650 C,
(b) and (d) are the error bounds for (a) and (c) respectively.
(B) MT-DATA prediction for (a) the amount of Laves phase and (b)
the amount of Mo in solid solution in austenite for a steel of
same composition (phases allowed were austenite,
, Laves phase, phase, ferrite and liquid).
The base composition used can be found in .
There are no specifications as to what the maximum level of
molybdenum should be for the AISI 304 steels. It is common to find up
to 0.5 wt% Mo in these steels. For AISI 316, an addition of 2-3 wt%
Mo is specified. It has been shown that molybdenum has a beneficial
effect on creep strength because of its solution strengthening role,
although this effect can disappear after prolonged ageing due to the
formation of Mo-rich Laves phase .
Figure 8 illustrates the predicted effect of
molybdenum on the 104 h rupture stress.
The initial increase is consistent with equation 2,
which predicts a strong effect of small additions of Mo in 304.
The predicted gradient (assuming a linear variation) is 38 MPa / wt%
between 0 and 0.02 Mo wt%, which is lower than the one given by
Particularly interesting is that the trend between 0 and 1.1 wt% of
molybdenum shows an excellent agreement (correlation 0.995) with
a c1/2 (where c is the concentration) dependence expected for
a solution strengthening mechanism .
The flattening of the curve would be consistent with precipitation of a
molybdenum rich phase, which would keep the matrix content at a
constant level. In this regard, the shift of the plateau between the
104 h and the 105 h rupture stress could be related to the
kinetics of this precipitation.
Calculations made on this composition with MT-DATA
(fig. 8B) actually reveal a consistent trend in
the Mo content of the austenite when increasing the bulk Mo content,
but indicate that Laves phase is only expected for Mo content greater
than 2.2 wt%.
According to equation 2, chromium slightly reduces
the creep rupture strength. Unfortunately the mechanism does not seem
to be understood.
It was possible to reproduce this trend for a type 316 steel as
illustrated in figure 9. The gradient for
compositions close to 16 wt% Cr is in very good agreement with the
value of 7.5 found in equation 2.
(a) The predicted influence of chromium on the
104 h rupture stress at 650 C for a typical 316.
(b) The effect of boron on the creep rupture stress for
two different steels. Detailed compositions for these examples are
given in table 2
Additions of boron have been found to increase substantially the
creep life of austenitic stainless steels, as emphasised by the large
coefficient it is attributed in equation 1. The
predicted effect of boron was found to be similar, with a slope
of about 5700 MPa / wt% (see figure 9).
There has been much work on the influence of the amount of `stabilising
elements' such as Nb, Ti, V or Zr, which prevent the formation of
chromium carbides, on the creep properties of austenitic
stainless steels. The problem is generally described using a
stabilisation ratio (eq.8),
which is a convenient estimate of the extent to which carbon, nitrogen
and the stabilising elements deviate from stoichiometry during
compound formation (, , )
Keown and Pickering  estimated that the best
creep properties were obtained for stoichiometric additions of Nb
while more recent work  claims that `under stabilising' C
and C kept in solution is better.
This is because data from long term experiments  have
shown that the trend observed by Keown and Pickering for rather short term
experiments (average 3000 h) do not extrapolate well.
For Ti on its own, the agreement is that a larger stabilisation ratio
produces the optimum creep strength 
(a) The effect of the stabilisation
ratio (increase of Nb) on the creep rupture life of a typical
18Cr-12Ni steel at 650 C. Optimum short term creep properties are
obtained for close to stoichiometric additions,
while understabilisation is better for long term properties.
(b) the effect of an increase of Ti: the stabilisation ratio which
produces the optimum creep strength is larger than in (a).
For short term tests, optimum creep properties have often been found
when the precipitation of MX was maximised by using stabilisation
equal to or greater that unity. However, according to ,
the creep of austenitic steels is essentially diffusion controlled in
service conditions, which could explain why there is no great benefit
in maximising the amount of MX precipitates.
Figure 10(a), shows that the neural network model was able to
reproduce the expected trends for Nb additions.
For short term tests, the best properties are obtained for a stabilisation
ratio equal to or slightly greater than unity.
For longer term tests, the influence of stabilising elements is predicted to
be less and the optimum addition much below a stoichiometric ratio.
On the right of the same figure is the effect of Ti addition on
the 104 and 105 h rupture stress of a typical 18-12 steel, showing that
the model correctly predicts best properties at a larger stabilisation
ratio, although this ratio still shifts slightly towards
understabilisation for longer times.
(a) the predicted effect of the solution
treatment temperature on the creep rupture life of a typical 18-12
steel (see table 2).
On the right, the amount of NbC present as a function of
the solution treatment temperature, in the 0.32Nb wt%, calculated
with MT-DATA (allowing for austenite, ferrite, liquid, TiC, NbC, TiN, NbN,
and phase). The solubility limit corresponds closely to that
for which optimum creep properties are predicted
Figure 11 (a), shows the effect of the solution
treatment temperature for different levels of Nb (see base composition
in table 2). This is in very good agreement with
the hypothesis  that the optimum creep properties are
obtained when as much as possible of MX forming elements are put in
solution before service: with an increased level of niobium, the
solution treatment temperature that dissolves the maximum amount of
Nb(C,N) is increased.
Figure 11(b), is a prediction of the amount of NbC
(calculated with MT-DATA ) found in a
steel of composition equal to that used for the predictions above,
(steel with 0.32 Nb wt%). It shows that all of the carbon and niobium are in
solution only at temperatures larger than 1250 C, which closely
matches the optimum solution treatment temperature for this composition.
The recent revision of the NRIM (National Research Institute for
Metals, Japan) datasheet no. 28 (28B, data for SUS
347H TB) contains considerably more long term data than did
the previous version 28A.
At the time when the database used in the present work was compiled,
these new data were not available and therefore have not been used to
train the models. It is interesting to compare the predictions of our
model with those made by the NRIM on the same data.
The 105 h rupture stress was predicted using the neural network
model for two steels of the NRIM 28B datasheet (AEA and AEG), for
temperatures ranging between 600 and 750 C.
Comparison between the neural network model
predictions (shaded area), the predictions made by NRIM using the
Orr-Sherby-Dorn method (line) and the experimental values published in
the recent revision 28B (points). See table 2 for full
Figure 12 shows the predictions of the neural network
model against those made by the NRIM, using the Orr-Sherby-Dorn
method, on the previously published data, and the recent results
published in revision 28B. The agreement with
experimental data is good, particularly in
the case of steel AEG where the neural network produces significantly
better predictions than the Orr-Sherby-Dorn method used by the NRIM. It
should also be noticed that the trends for both steels have been
correctly predicted, despite their apparent similarity in composition
(see table 2).
The base compositions of the different examples, in wt%.
A number of other trends predicted by the models have been examined,
which have been found to be reasonable from a metallurgical point of
view, amongst which a positive effect of Cu, of P within a limited range
(however this element causes embrittlement and is therefore kept much
below the level giving optimum creep rupture strength). The
number of possibilities of interactions are such that it is not
possible to study them fully.
The database used to create the model covers a large range of
compositions and its application does not stop at the AISI 300 series.
The software capable of doing these calculations can be obtained
freely from http://www.msm.cam.ac.uk/map/map.html.
The creep rupture life for a given stress, and the creep rupture
stress for a given life have been analysed using a neural networks
method within a Bayesian framework. The data were obtained from a
variety of sources and
cover a wide range of compositions and heat treatments.
The potential of the method is clearly illustrated in its ability to
perceive interactions between the different input variables. Predicted
trends have been found consistent with those expected and the
quantitative agreement was frequently satisfying. The model can be applied
widely because of its capacity to indicate uncertainty, including both
an estimate of the perceived level of noise in the output, and
an uncertainty associated with fitting the function in the local
region of input space.
The authors are grateful to Innogy Plc and EPSRC for funding the
project of which this work is part, and to Pr. D. Fray for provision
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