Abstract
Though the basic science behind bake hardening has been known for some time, designing alloys is difficult in that the final bake hardening property depends mainly on the aging conditions and the amount of free C, which itself is a function of alloying additions, cooling rates, etc. In the present work, the amount of free C along with aging condition and deformation was measured before aging and taken into consideration to make a model which was used to resolve certain issues related to their effects on the final strength increment.
Introduction
Bake hardening (BH) steels are popular with the automobile industry for their excellent formability, coupled with some extra strength when all the forming operations are over and the parts are being painted (Fig. 1). Though the essence of BH is strain aging and has been known for a long time, in recent work, Das et al.1 showed the dependence of BH properties on many variables which include chemical composition, and prestraining and aging conditions. However, it was not possible in that case to correlate the effect of total or free C with the final strength increment due to bake hardening Δσ even though this quantity is considered as one of the major influencing parameters in determining Δσ. Here an attempt was made to build a simpler model with fewer variables through which the effect of free C, aging condition and prestraining on final Δσ can be evaluated.
Definition of bake hardening
Technique
A good method for recognising patterns in complex data is the neural network in a Bayesian framework and has been used extensively.1 – 11 For this reason, only specific points of relevance are introduced here and the details can be available from a recent thesis.11
The network is a non‐linear regression method which, because of its flexibility, is able to capture enormous complexity in the data and at the same time, avoids overfitting. There are a number of interesting outputs other than the coefficients which help to recognise the significance of each input. First, there is noise in the output, associated with the fact that the input set is unlikely to be comprehensive, i.e. a different result is obtained from identical experiments. Second, there is the uncertainty of modelling because many mathematical functions may be able to adequately indicate known data, which behave differently when extrapolated. Knowledge of this uncertainty helps to make the method less worrisome in extrapolation. This is also important in identifying domains where care must be exercised in interpreting the results from a network. Finally, there is the significance of each input in explaining variation in the output, akin to a partial correlation coefficient in multiple linear regressions. The significance is dimensionless because the variables are all normalised within ±0·5 for the purposes of creating the neural network model.
Data
All the raw data were collected from many sources.12 – 19 The bake hardening effect is the result of the migration of the interstitial solute atoms to the dislocation sites during the paint baking cycle. Keeping this in mind, an attempt here was made to correlate the amount of interstitial elements with the final bake hardening properties. In the current work, the reported values of free C in the respective papers or theses were only considered. It should be kept in mind, however, that in almost all the cases considered, free C was measured by an internal friction technique which cannot detect the entire free C content20 and hence gives a somewhat lower value and this can be manifest in the noise in the model.
Bake hardening of ultra low carbon steel is effectively a strain aging phenomenon. To consider the kinetics of strain aging, ‘Dt’ was used for training where D = 5·2×10−4exp (−9000/T) cm2 s−1, the diffusion coefficient of carbon in ferrite, T is the aging temperature in K and t is the aging time in s. Using Dt over aging time and temperature has got two advantages: first, it gives a kinetic view of the aging process and second, it helps to reduce the process variables. Directly using Dt would have created some confusions in the model as it would have faced some problems during the normalisation of the input parameters. To avoid this the natural logarithm of Dt, i.e. ln (Dt) was used instead of Dt. Unlike the previous paper,1 here prestraining at elevated temperature was not considered. In Table 1, a list of the input variables with their maximum, minimum and average values with the standard deviation is provided.
Properties of data used in creating model
The extent of agreement between the predicted and the measured values of bake hardening is illustrated in Fig. 2. The plot represents the entire available dataset of some 580 experiments with the calculations performed using the optimised committee of models as described elsewhere.3 – 6 The error bars represent ±1σ modelling uncertainty.
Extent of agreement between predicted and measured values of bake hardening: plot represents entire available dataset of some 580 experiments; error bars represent ±1σ modelling uncertainty
Application of model
Four different cases were selected in order to explore the behaviour of the model. The parameters (free carbon in ferrite, amount of prestrain and Dt) for each of the steel are different and listed in Table 2 and form the basis for studying the trends. It may henceforth be assumed that when a plot of Δσ is presented against a particular variable (e.g. prestrain), then the values of all the other variables correspond to those listed in Table 2.
Examples of steels studied using model
Aging condition
The BH effect in ultralow carbon steel is effectively an accelerated strain aging which can depend on the aging time and temperature. In the present analysis, Dt is used to take care of both these parameters. The effect of Dt on Δσ can be seen from the Fig. 3. In Fig. 3a , Δσ is plotted as a function of Dt for steel 4 whereas Fig. 3b shows the dependence of Δσ on Dt for all the four steels. Two distinct stages of strength increment can be observed from both the figures. The first one, as indicated in the literature,12 – 14,17 is due to the formation of Cottrell atmospheres, and the second increment is due to the precipitation of extremely fine carbides.12 – 14,17 The higher bake hardening response at higher Dt value can be due to the possible formation of ϵ‐carbide on the existing dislocations.21 With increasing Dt, more C atoms come to the dislocation sites and, as a result, strength increment occurs due to the formation of Cottrell atmospheres. With continuously increasing Dt, as pipe diffusion along the dislocation line continuously takes place, clusters of C atoms begin to form on the dislocation sites.22 Eventually, nucleation of very fine precipitates occurs from these clusters. 21 21,22 Although the existence of this kind of precipitates was not observed using transmission electron microscopy (TEM), the work by De et al.14 confirmed the formation of low temperature carbides at the later stages of bake hardening.
Calculated Δσ as function Dt for a steel 4 and b steels 1–4
It can be seen from Fig. 3b that the bake hardening effect does not increase continuously with increasing Dt, rather, it reaches a plateau. This behaviour is expected because each steel, before the baking treatment, is having a particular amount of free carbon which gets consumed first due to the Cottrell atmosphere formation, and second due to precipitation. Therefore, after a certain time, no free carbon atoms are available there in the steel to contribute to the bake hardening response and it becomes insensitive to Dt.
Free carbon
In a previous study,1 it was not possible to correlate the effect of the amount of total or the free carbon content with the bake hardening effect. In the current analysis, it can be seen from Fig. 4 that bake hardening response initially increases with the amount of free carbon, but then it effectively reaches a saturation point and then again increases with increasing free carbon content. It shows that at a given Dt, carbon atoms first form the Cottrell atmosphere and then with increasing free carbon content, precipitates occur. From this figure, it can be said that effective bake hardening is obtained when the amount of free carbon remains at ∼10 ppm which is in accordance with previous work that were not been included in the dataset for neural network analysis.23 – 25
Calculated Δσ as function of amount of free carbon in ferrite for steel 3
In order to effectively increase the strength due to bake hardening treatment, C atoms first need to go to the dislocation sites and subsequently precipitate will form. Both of these stages need energy and that energy can only come from the aging treatment. Thus the effect of free C at a fixed aging condition (i.e. Dt) reflects the situation only partially and the combined effect of free C with different aging conditions should be taken into consideration. Accordingly, an effort in the current study was also made to determine the combined effect of free carbon in ferrite and Dt on the final Δσ property and the results are shown for steel 2 in Fig. 5 where a three‐dimensional contour plot is presented having free C as its X axis, ln (Dt) as Y axis. The predicted BH values and the corresponding uncertainties are plotted as different colours and isocontour lines respectively in MPa.
Combined effects of Dt and free carbon for steel with 5% prestraining: isocontour lines represent Δσ, whereas different colours are for different levels of uncertainties; units of numbers mentioned on isocontour lines and on different colour codes are in MPa
From Fig. 5 it can be seen that a higher bake hardening effect can be obtained at a combination of high Dt with higher amount of free carbon. However, using this combination to get a higher strength comes with high degree of uncertainty in the prediction. For steel with ∼10 ppm free carbon and high Dt, the model predicts a reasonably high degree of bake hardening response at an acceptable range of uncertainty in the prediction. This finding is also in accordance with the experimental values reported in the literatures.23 – 25
Prestrain
In most of the applications of steels having bake hardening behaviour, the workpiece undergoes a small amount of straining during the final forming operation before the baking process. The bake hardening experiments are, therefore, carried out with samples with a small amount of prestrain. However, there is controversy about the effect of prestrain on bake hardening response, for instance, some researchers found an initial increase followed by decrease in strength increment due to baking with increasing prestraining,15 whereas others found a continuous decrease in bake hardening response with increasing prestrain,12 and some also reported that bake hardening effect does not depend on the amount of prestrain at all. 14 14,16
In the current analysis, it can be seen from Fig. 6 that increasing prestraining does not affect the final bake hardening property positively, rather, over prestraining may even lead to a reduction in the final BH response. For any particular steel the amount of free carbon is fixed for a given set of processing parameters. Therefore, increasing the amount of deformation can only increase the dislocation density which may not be locked by the existing amount of free carbon. Hence it will be interesting to check the simultaneous effect of increasing both the amount of prestraining and the amount of free carbon for a particular aging condition. Steel 2 was chosen for this analysis and the corresponding results are shown in Fig. 7 where X and Y axes are represented by the amount of free C in ppm and percentage of prestraining respectively. Corresponding BH response in MPa is shown as different colour codes in the background and the isocontour lines, representing different levels of uncertainties (in MPa) in the prediction, are superimposed on the predicted results.
Calculated effect of prestraining on bake hardening property for different steels
Calculated effect of prestrain and free carbon for steel 2: contour lines show different iso‐BH profiles and different colours represent varying degrees of uncertainties in predictions; units of numbers mentioned on isocontour lines and on different colour codes are in MPa
Figure 7 clearly shows that with increasing prestrain, final BH response decreases for a particular aging condition. Such a reduction might be expected if not all the dislocations introduced during prestraining become pinned during the aging process. A higher predeformation would leave higher amount of free dislocations and hence a diminished bake hardening response could be the outcome. This hypothesis will only be valid if the aging condition is insufficient and hence many C atoms might not be able to migrate to the dislocation sites to lock them up. If sufficient time and temperatures were given, the C atoms might have been able to travel to the all possible locations of dislocation sites and lock them.
To cross‐check this hypothesis, another analysis was carried out where the amount of prestraining and aging condition (i.e. Dt) were varied keeping the initial amount of free carbon constant. The results are shown in Fig. 8 from which it can be seen that given sufficient time and temperature, all the dislocations may be locked and the total bake hardening response does not depend very much on higher amount of prestraining provided that there are enough amounts of carbon atoms to lock them up. This kind of behaviour was also reported earlier.1
Calculated combined effect of increasing amount of prestraining and Dt on final bake hardening effect for steel with 8 ppm free C: units of numbers mentioned on isocontour lines are in MPa
Validation of model
To validate the model one steel was chosen to carry out the bake hardening experiments. The composition of the steel is given in Table 3. This steel had ∼5 ppm C (as measured by internal friction), prestrained to 2% and then bake hardened at different temperatures from 170 to 225°C for 20 min. The corresponding model predictions and the experimentally obtained bake hardening values are mentioned in Table 3 which shows an excellent match between the predictions and final results.
Validation of model: predictions as well as experimentally measured values of BH are for steel containing 5 ppm C and prestrained to 2%
Conclusions
A neural network model has been developed considering the measured amount of free C, aging condition and predeformation. The model behaves quite well and predicts the final properties which are logical and consistent with basic understanding. With the help of neural network, it has now been possible to rationalise the effects of free C content on the final bake hardening property.
Footnotes
Acknowledgements
S. Das is grateful to Commonwealth Scholarship Commission (UK) for financial support towards his study in Cambridge University during 2006–2007 and to Professor H. K. D. H. Bhadeshia for his generous guidance and fruitful discussions.