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 techniques.

Introduction

Austenitic stainless steels are commonly used in the power generation industry at temperatures greater than 650 $^\circ$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 ( $\sigma_{f, 10^{4}}$) of a AISI 316 at 650 $^\circ$C,

$$\sigma_{f, 10^{4}\mathrm{h}}=173.8+7243(\mathrm{B})+961.1( \mathrm{N})+1145(\mathrm{S}) -7.5(\mathrm{Cr})$$ (1)

while equation 2 is for a 304 steel at the same temperature [1]:

$$\sigma_{f, 10^{4}\mathrm{h}}=90.81+115(\mathrm{Mo}) +498.5(\mathrm{W})$$ (2)

where the concentrations are in wt%. The range for each variable is not given in [1], 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.

Method

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.

Figure: 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.

$$h_{i}=\tanh \left( \sum_{j} w_{ij}^{(1)}x_{j}\ +\ \theta^{(1)}_{i} \right)$$ (3)

$$y=\sum_{i}w_{i}^{(2)}h_{i}\ +\ \theta^{(2)}$$ (4)

x_i are the inputs and w_i the parameters, or weights, which define the network. The biases $\theta_{i}$ 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 $\pm 1$, 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:

$$\beta E_{D} + \alpha E_{W}$$ M(w) =

$$\frac{1}{2}\sum_{i}\left( t^{(i)}-y^{(i)} \right)^{2}$$ E_D =

$$\frac{1}{2}\sum_{i} w_{i}^{2}$$ E_W = (5)

where E_D is the overall error, and E_W the regulariser, used to force the network to use small weights (equations 5). $\alpha$ and $\beta$ are control parameters which largely influence the complexity of the model. t⁽ⁱ⁾ is the target for the set of inputs ${\bf x}^{(i)}$, while y⁽ⁱ⁾ is the corresponding network output.

The method used in this study, developed by MacKay [4], is based on Bayesian probability theory and treats learning as an inference problem.

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 exceptionally noisy.

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.

$$\mathrm{LPE}=\sum_{m} \left[ \frac{1}{2} \left( t^{(m)}-y^{(... ...} + \log \left( \sqrt{2\pi}\ \sigma_{y}^{(m)} \right) \right]$$ (6)

where $\sigma_{y}^{(m)}$ is related to the uncertainty of fitting for the set of inputs ${\bf x}^{(m)}$.

The overfitting problem

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 $\alpha$ and $\beta$ 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.

Figure: (A) When a model has overfitted the training data ($\bullet$), 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.

Figure 2 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.

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 $\pm 0.5$, this step is not obligatory but is a convenient way of comparing the effect of different variables on the output.

Committee Model

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, $\bar{y}$, of a committee of networks is the average prediction of its members, and the associated error bar is calculated according to:

$$\bar{y} \frac{1}{L}\sum_{l}y^{(l)}$$ =

$$\sigma^{2} \frac{1}{L} \sum_{l} {\sigma_{y}^{(l)}}^{2} \ +\ \frac{1}{L} \sum_{l} \left( y^{(l)}-\bar{y} \right)^{2}$$ = (7)

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 prediction y^(l). 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 model. The selected models are then retrained on the full database.

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 testing sets.

Figure 3 gives an idea of the distribution of each input against the logarithm of the rupture life.

Figure: 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 documented interactions.

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 [5] 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 [22], 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.

Table 1: The different inputs in the data set.

Input variable	Minimum	Maximum	Mean	Std deviation
Test Stress (MPa)	5	443	145	72
Test Temperature (oC)	500	1050	667	71
log (rupture life / h)	-0.200	5.240	3.324	0.879
Cr wt%	12.98	22.22	18.08	1.35
Ni wt%	8.40	32.48	13.82	5.47
Mo wt%	0.00	2.82	1.05	1.10
Mn wt%	0.56	2.50	1.36	0.35
Si wt%	0.040	1.150	0.545	0.171
Nb wt%	0.000	2.980	0.242	0.449
Ti wt%	0.000	0.560	0.131	0.199
V wt%	0.000	0.090	0.004	0.011
Cu wt%	0.000	0.310	0.051	0.074
N wt%	0.000	0.170	0.029	0.052
C wt%	0.012	0.330	0.062	0.025
B wt%	0.000	0.005	0.001	0.0016
P wt%	0.000	0.038	0.021	0.0067
S wt%	0.000	0.030	0.012	0.0071
Co wt%	0.000	0.540	0.037	0.1090
Al wt%	0.000	0.520	0.029	0.0804
Solution treatment temperature (oC)	1000	1350	1102	51

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 [1].

Creep rupture life model

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:

$$\mathrm{Stabilisation\ Ratio}=\frac{\mathrm{Nb}/8+\mathrm{Ti}/4}{\mathrm{C} + \mathrm{N}}$$ (8)

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 .

Figure: The perceived level of noise $\sigma_{\nu}$ (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 $\sigma_{w}$ 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.

Figure: The perceived significances $\sigma_{w}$ 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.

Figure: The performances of the final committee model on the whole database, for the rupture life model (a) and the creep strength model (b).

Creep strength model

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 database.

Figure: 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.

Applications

Molybdenum in AISI 304 and AISI 316

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 $^\circ$C are respectively around 80 MPa and 110 MPa [23].

Figure: (A) The predicted effect of Mo on (a) the 10⁴ h and (c) the 10⁵ h rupture stress at 650 $^\circ$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, ${\rm M}_{23}{\rm C}_{6}$, ${\rm M}_{6}{\rm C}$, Laves phase, $\sigma$ 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 [1].

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 equation 2. Particularly interesting is that the trend between 0 and 1.1 wt% of molybdenum shows an excellent agreement (correlation 0.995) with a c^1/2 (where c is the concentration) dependence expected for a solution strengthening mechanism [24].

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%.

Chromium and boron

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.

Figure: (a) The predicted influence of chromium on the 10⁴ h rupture stress at 650 $^\circ$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).

The stabilisation ratio and solution temperature

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 ([25], [13], [26])

Keown and Pickering [13] estimated that the best creep properties were obtained for stoichiometric additions of Nb while more recent work [27] claims that `under stabilising' C and C kept in solution is better. This is because data from long term experiments [27] 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 [1]

Figure: (a) The effect of the stabilisation ratio (increase of Nb) on the creep rupture life of a typical 18Cr-12Ni steel at 650 $^\circ$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 [1], 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.

Figure: (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, ${\rm M}_{23}{\rm C}_{6}$ and $\sigma$ 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 [25] 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 [28]) 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 $^\circ$C, which closely matches the optimum solution treatment temperature for this composition.

Comparison with other methods

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 $^\circ$C.

Figure: 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 compositions.

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).

Table 2: The base compositions of the different examples, in wt%.

Fig.	Cr	Ni	Mo	Mn	Si	Nb	Ti	V	Cu
8	18.0	12.0	-	1.4	0.6	0	0	0	0
9(a)	-	12.1	2.54	1.41	0.46	0	0	0	0
9(b) 316	16.42	13.21	2.34	1.51	0.52	0.01	0.011	0	0.14
9(b) 347	17.89	12.55	0.11	1.74	0.77	0.77	0.02	0.033	0.09
10(a)	18.15	13.3	0	0.75	0.4	0.1	0	0	0
10(b)	17.71	12.27	0.02	1.56	0.55	0.005	-	0	0.06
11(a)	18	12	0.05	0.8	0.4	0.02	0.02	0	0
12(AEA)	17.85	12	0.04	1.71	0.60	0.74	0.019	0.031	0.05
12(AEG)	17.56	12.24	0.15	1.81	0.63	0.87	0.019	0.041	0.14
Fig.	N	C	B ppm	P	S	Co	Al
8	0	0.06	5	0.02	0.01	0	0
9(a)	0	0.04	0.0002	0.019	0.02	0	0
9(b) 316	0.034	0.05	-	0.021	0.01	0	0
9(b) 347	0.016	0.05	-	0.025	0.007	0.37	0.004
10(a)	0.012	0.062	0	0.02	0.002	0	0
10(b)	0.014	0.06	5	0.026	0.01	0	0.121
11(a)	0.01	0.06	10	0.02	0.002	0	0
12(AEA)	0.0284	0.07	12	0.02	0.005	0.29	0.019
12(AEG)	0.0222	0.053	27	0.027	0.011	0.30	0.008

Software

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.

Summary and conclusions

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.

Acknowledgements

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 of laboratory facilities.

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