Training the Model

The data were randomly and equally divided into the training and testing sets, and normalised [11]. One hundred networks were trained, with hidden units ranging from one to twenty and five seeds in each case. This is in order to select a committee of models which gives the best generalisation on unseen data [29,14,13,11]. The performance of the optimum committee accompanies by $\pm 1 \sigma$ modelling uncertainties is illustrated in Fig. 2. Of the total of 12807 data, only 158 can be classified as mild outliers which are more than 3$\sigma$ from the measured values. The noise in the output of the committee model was found to be $\sigma= \pm 4\%$, which is a constant additional error to the modelling uncertainties plotted in subsequent graphs. The network perceived significances, which indicate the ability of an input to explain the variation in the output (akin a partial correlation coefficient) are shown in Fig. 3. The elongation, ultimate tensile strength and proof stress are significant in influencing $da/dN$ but it is natural that the stress intensity range $\Delta K$ should have the greatest effect. Although it is expected in a valid test that specimen size should not influence $da/dN$ [30], it is likely that true plane strain conditions do not exist in all the cases studied, and hence a specimen size effect is perceived in Fig. 3. Such behaviour has been reported previously, with the crack growth rate increasing as plane-strain conditions are approached [31].

One way of assessing a model is by making predictions, in this case on a bearing steel of relevance in our other research. The steel of interest is variously known as SUJ2, AISI 52100 and En31 in different countries and has the approximate composition 1C, 0.3-1.1Mn, 1.2-1.4Cr, 0.2-0.4Si wt%. The inputs required were obtained from [32]: 5% elongation, 2030 MPa 0.2% proof stress, 2240 MPa tensile strength, loading mode 2, specimen length 80 mm, specimen thickness 2 mm, pre-crack size 3 mm, frequency 2 Hz and stress ratio 0.

Fig. 4 shows the outcome, with the model not only capturing the trend in the variation of $da/dN$ versus $\Delta K$ over several orders of magnitude, and both for the threshold and Paris regions of the curve, but giving also a reasonable absolute prediction accuracy.

Although all of the data used to create the model were from experiments on steels [25], the inputs include only mechanical and test parameters. It was imagined that the model should therefore apply without modification to other alloys.