Predicting Strain-Induced Martensite in Austenitic Steels by Combining Physical Modelling and Machine Learning

Computational materials design has made significant progress lately. However, one underexploited opportunity lies in the combination of physically based modelling and machine learning (ML). In the present work we exploit this combination for modelling of strain-induced martensitic phase transformation (SIMT) in austenitic steels. A fully predictive model for SIMT, responsible for the TRIP effect in many steels, is devised. An experimental dataset correlating SIMT with composition, temperature and strain is collected from the open literature firstly. Secondly, the Olson-Cohen model is applied to make physically based predictions on temperature and strain dependence of SIMT in order to expand the database to the final size of 16,500 entries relating the features and the target. Thirdly, ensemble ML methods are applied to model the data and the final model is validated on a holdout dataset, including also dual-phase alloys. The final model provides accurate predictions of SIMT in a temperature range from −196 to 100 °C and from 0 to 1 in strain. The model can readily be extended to consider further factors such as strain rate and stress state. Moreover, it can be used together with thermodynamic and kinetic calculations, or thermomechanical simulations, for the design of steels and components, respectively.


H I G H L I G H T S
• An integrated machine learning and physical modelling to predict straininduced martensite in austenitic steels. • A fully predictive model of SIM features, responsible for the TRIP effect in many steels, is devised. • The model provides accurate predictions in comprehensive temperature and strain range in austenitic and dual phase steels. • The model can readily be extended to consider further factors such as strain rate and stress state.

G R A P H I C A L A B S T R A C T
a b s t r a c t a r t i c l e i n f o

Introduction
Advanced metallic materials are imperative to meet current grand challenges. For example, there are urgent needs for: high-strength materials enabling light-weight design; durable alloys prolonging service life by avoiding degradation; and, metallic materials with excellent Materials and Design 197 (2021) 109199 high-cycle fatigue resistance facilitating electrification of transportation. Austenitic stainless steel is one alloy category that has bearing on all the mentioned applications with its attractive tunable mechanical properties and high corrosion resistance. The mechanical properties can be controlled by alloying and processing to achieve either high strength or a rather low yield limit but with exceptional strain-hardening and ductility [1]. The tunable properties are due to the strain-induced martensitic transformation (SIMT) and control of SIMT is thus critical to optimize mechanical properties for different applications [2,3]. It is, for example, possible to design for optimum formability at different temperatures or excellent crash absorption considering stress state and strain rate [4].
Following the current trend of the materials community with increasing computational materials design ambitions it becomes critical to be able to model the important SIMT. SIMT is not only important in austenitic stainless steels, but also numerous other steels benefit from SIMT and the TRIP effect [1]. In this work we address SIMT in austenitic stainless steels to assess whether combining physically based modelling and machine learning can successfully be used as part of a computational materials design framework enabling optimization of SIMT in steels.
The most popular physically based model for SIMT is the Olson-Cohen (O-C) model [5]. Using the O-C model the SIM fraction can be modelled with respect to the applied strain using model parameters related to thermodynamic driving force and stacking fault energy. The original O-C model has been used frequently and extensions to the original model have also been proposed. For example, the effect of strain rate has been included [6,7]. These models have subsequently been applied in finite-element simulations to predict SIMT during e.g. mechanical testing [2] and real forming operations [8]. There are also other physically based models for SIMT [9]. However, all these models, including the O-C model, require fitting to experimental data before they can be applied on other similar materials and test conditions. A fully predictive model where only the chemical composition and test conditions are given to simulate the stability of austenite in e.g. forming operations would be very useful.
Machine learning (ML) technology has a proven track-record in predicting various materials properties [10,11]. In fact, ML, specifically artificial neural networks (ANN), has already been applied to model SIMT in austenitic stainless steels [12,13]. However, the model predictions in those works were quite unsatisfactory with error margins of the predicted fraction of SIM of about ±0.2. Still, those works show the feasibility of applying ML for SIMT and if some of the deficiencies can be solved it is believed that an accurate predictive model could be developed. We have identified three main potential improvements from prior works: i) for accurate and wide applicability of the ML model a sufficiently large database must be collected; ii) the recent advance in ML algorithms that are more effective to treat small datasets as compared to ANN [14] should be used; iii) a continuous feature like SIMT fraction, often just determined by relative measurements to the original state of the material, is not perfectly suited for statistical modelling by standard ML, but the dependence of SIMT fraction on applied strain and temperature can be well described by physically based models; thus, this should be exploited.
In the present work we therefore apply a combinatorial approach, where physically based modelling using the O-C model is combined with state-of-the-art ML modelling, to model SIMT. A fully predictive model considering the effect of temperature, strain and alloy composition to predict the SIM fraction in austenitic stainless steel and related alloys is devised.

Combinatorial modelling methodology
The combinatorial modelling methodology pipeline, combining the fields of statistics and physics, is presented in Fig. 1. The pipeline starts from the left-hand side with database development and thereafter physical modelling to expand the database. Feature engineering, including feature selection, scaling and discretization, overlaps the first two steps. ML is the next step in the pipeline and the final step is the analysis and extraction of important insights. The different steps of the modelling pipeline are treated more in detail in sections 2.1 (Database development), 2.2 (Physical modelling and database expansion) and 2.3 (ML); the discussion of insights gained from this modelling work in section 3.3, follows after the presentation of the results in sections 3.1 and 3.2.

Database development
Experimental data of SIMT was collected from the open literature [2][3][4]9,. Data for the two main types of austenitic stainless steels, i.e. AISI 200-series (Mn alloyed) and AISI 300-series (Ni alloyed) as well as related alloys, were included in the data collection. The database is summarized in Table 1. Only data from slow strain rate (≤ 0.001 s −1 ) uniaxial tensile testing were included in the database. It is well-known that both the strain rate and the stress state will influence the transformation but as a first step, to reduce the complexity of the model, it was decided to exclude these parameters here. Hence, only slow strain rate tests were included where temperature is kept almost constant and the minor differences arising due to small changes in strain rate can be neglected. It can be noted that the availability of data on strain rate and stress state is not large in the literature and, hence, it may be most suitable to add these effects through a physical model at a later stage.
The feature selection process was aided by first evaluating the linear correlation of the features and the target, so-called Pearson correlation, see Fig. 2. It was clear from this analysis that the elements C, N, Cr, Ni, Mn, Mo, Si, Cu, Co and Nb have a non-marginal influence on SIMT. Therefore, all these elements were included as features. In addition, it was also indicated that the two most important features were temperature and strain. Hence, the database expansion using the physical model was applied in the next step (section 2.2) to expand the database with respect to these two important parameters. Prior to that, data cleaning was applied on the collected database to make sure that poor data entries and errors in the original data were corrected. In addition, it was also found that certain elemental compositions were not included in the original data, especially values for N compositions. In those cases, a standard N composition value for that specific grade was filled in to the empty data entry position in the database. Considering the similarity of the scales of features in the database it was found that no scaling was necessary. It could be noted though that Fe was not included as an explicit feature, since it was evaluated to contribute in a negative way to the accuracy of the ML modelling.

Physically based modelling and database expansion
The O-C model [5] was applied to model the collected experimental data and to expand the database with additional data for the strain and temperature dependence of SIMT. It should be noted that the O-C modelling was applied for one alloy at the time, i.e. the data for one alloy was fitted with respect to strain and temperature, and hence, it should not be confused with the predictive modelling that was enabled through ML (section 2.3). The O-C model is based on the assumption that the formation of martensitic embryos is directly proportional to the number of shear-band intersections. This assumption is fair, but it should also be mentioned that prior work has indicated that the number of shear-bands is more important than the shear-band intersections for highly unstable austenite [50]. The governing equation for the O-C model is: where α relates to the probability of shear-band intersection formation; β relates to the probability for the formation of a martensite embryo at a shear-band intersection; and, the exponent n is frequently a constant of 4.5 [4], which has been shown to give appropriate fitting for a range of austenitic steels. The shear-band formation, and the intersecting of shear-bands (α), are related to the stacking fault energy (SFE) of the alloy and the applied strain rate. This also means that α is temperature dependent, since the SFE depends on temperature [51]. β is also temperature dependent, but in this case because the driving force for BCC-martensite formation depends on temperature [5].
An example of the physically based modelling is shown in Fig. 3. Experimental data for austenitic stainless steel 304 [3] and 301LN [39] that were uniaxial tensile tested at different temperatures is modelled by the O-C model. It can be seen that the experimental SIM fraction is well described by the O-C model when using appropriate α and β parameters.
The O-C model is fitted to the experimental data from 0 to 1 strain for all experimental datasets, which means that a continuous function, with interpolated and extrapolated strain values relating to the SIM fraction, is generated between 0 and 1 strain. To further expand the database with respect to temperature, the O-C model is also used to interpolate to temperatures in-between the temperatures in the experimental database. This is performed by first evaluating α and β by modelling the experimental data at known temperatures for a specific steel grade (α T and β T ). Thereafter, these two parameters are used to find the parameters for a new temperature (α Tu and β Tu ) by linear fitting. One example of the modelling of the temperature dependence of SIMT is shown in Fig. 3 where the SIM fractions for grade 304 is modelled at 10 and − 30°C (dashed line) after fitting of the experimental data at −188, 0 and 30°C for the same alloy. After the application of the O-C model the experimental raw data is expanded to a larger continuous dataset for the SIM fraction with respect to the 12 features mentioned in section 2.1.
The continuous dataset must be discretized prior to application in ML, and after discretization the total database contains 16,500 data points. Fig. 4 shows the distribution of each feature in the database, excluding strain since all entries were modelled from strain 0 to 1.

Machine learning
A predictive model is now developed using ML on the expanded database. A prior evaluation has shown that for the type of database to be modelled here, ensemble methods may be preferred over ANN [14]. Therefore, the following methods were evaluated: Random forest (RF), Extremely randomized trees (ExT), Gradient boosting (GB) and Adaboost (AdB), where RF and ExT are bagging type ensemble methods whilst GB and AdB belong to the category of boosting type ensemble methods [52][53][54][55]. In general, boosting methods may be more susceptible to overfitting of the data [52,53]. A train-test split approach was applied where the dataset was randomly separated into a training dataset (80% of the data) and a test dataset (20% of the data). Prior to the modelling, tuning of the hyper-parameters was, due to the nature of the data, performed by manual iterations on the dataset to find the best compromise of bias and variance. The key parameters used for the modelling were: the number of trees in each forest (=600), the maximum depth of each tree in the forest (=20), and the maximum number of features considered for each split of the tree (=12). In the case of AdB, the base estimator was chosen to be decision tree throughout the modelling.  After finalizing the model and verification of the test predictions we also applied a second unseen (holdout) validation dataset that included other types of data, e.g. from duplex stainless steel in order to try and make sure that the generalization of the model was high. Using the model, it is possible to evaluate two predictors: SIM fraction given the chemical composition, strain and temperature, but also the Md30 temperature (SIM fraction 0.5 at 0.3 true strain), which is a common parameter when comparing the stability of the austenite in steels. The ML part of this paper was conducted using Python and the Scikit-learn library [56] and the database was administrated using the Python Data Analysis Library Pandas [57]. Fig. 5 shows an example of the predictions for the test data using the AdB model. The measured SIM fractions (x-axis) are compared with the SIM fraction predictions (y-axis) and the error band is set to ±0.05. It can be seen that more or less all test predictions fall within the error bands. The performance of the four different ensemble methods is very similar with RMSE values of about 0.01 for three of the methods (RF, ExT and AdB), whereas GB has a slightly higher RMSE of about 0.017. The AdB model was used as the final model. It should be mentioned that these error margins are exceptionally good and this is partly due to the procedure of applying a physically based model to fit the data and prepare a continuous function of strain data with respect to SIM fraction. Basically the testing here confirms that the ML can find the correlations in the data and represent the database appropriately. It does not, however, validate that the model generalizes well and is widely applicable for predictions. We will therefore perform a more independent analysis of the generalizability by validation on the holdout dataset in section 3.2.

Validation and generalization of SIM model
Predicted SIM fractions for the austenitic stainless steels are validated by further unseen data [58][59][60][61][62][63][64] to ensure that the final predictor model is generalizable for predictions in a wider range of alloys. The alloys included are austenitic stainless steel grades 301, 304 and 204 tested at temperatures from −100 to 80°C. Moreover, two duplex stainless steels (DSSs) with metastable austenite are also included in the validation set. The DSS grades, called FDX25 and FDX27, were tested  at room temperature using in-situ synchrotron x-ray diffraction during uniaxial tensile testing [65]. Their fraction of austenite prior to tensile testing was 58% for FDX25 and 64.4% for FDX27 [65], and the measured chemical compositions of the austenite was used as input for the ML predictor model. The validation results for grades 301, 304 and 204 are provided in Fig. 6 (a), (b) and (c) respectively. It should be noted that the N content of the alloys is not provided in the literature [58,60,62,64], and hence, a common N content of 0.02 wt% for 301, and 0.03 wt% for 304 was used for the input chemical composition. In general, the predicted SIM fractions are in good agreement with the experimental data for all grades.
The predictions for the 301 grade appears particularly accurate, which can be understood as an effect of the larger availability of data for this grade in the database used for the training of the ML model.
The SIM fraction evolution with increasing strain is even well described for the DSSs, see semi-solid squares and triangles in Fig. 6 (c). This is encouraging considering that there is no explicit DSS data in the current database. The uncertainty of predictions given by R 2 and RMSE values for all predictions as well as for each specific grade is summarized in Table 2. The R 2 for 301 series (Fig. 6(a)), 304 series ( Fig. 6  (b)), and, 204 series and DSS (Fig. 5(c)) is 0.974, 0.989 and 0.957, and RMSE is 0.0479, 0.0380 and 0.0369, respectively; whilst, R 2 and RMSE for the total validation dataset equal 0.979 and 0.0415, respectively. This is overall very good results and shows that the best predictions are obtained for the grades where data is most numerous, i.e. grades 301 and 304. Furthermore, the good predictions also for less prevalent data in the database such as for the 204 grade, and even the DSSs where no data is included in the database, hold promise for further development of a generic predictive SIM model. By incorporating more data and using a similar methodology as presented here it should be possible to develop a model for widespread implementation on metastable austenite in various steels such as duplex stainless steels, medium Mn steels, TRIP steels, etc.

SIM predictions interpreted using statistical and physical descriptions
As presented above, the developed predictor ML model can describe the SIM fraction as a function of the 12 features well, but we have not discussed the importance of each feature yet, i.e. which are the dominant parameters controlling the SIM fraction. To do this we present the evaluated feature importance from the predictor ML model in Fig. 7. Feature importance is evaluated by calculating the increase of the prediction error after permuting that feature [66]. The model error   will be increased due to the permutation if the feature is important, and the larger increase of the prediction error due to the permutation, the more important is the feature. On the contrary, a feature is evaluated to be unimportant if permuting that feature leaves the prediction error unchanged. The feature importance concept can thus be easily understood by considering that the model relies heavily on a certain feature to make accurate predictions. It can be seen in Fig. 7 that strain and temperature are the two most important features affecting the SIM fraction. After these two processing parameters, the importance of the elements starts with Ni and thereafter Si, N, C and Cr. The importance of the elements can now be compared with the two underlying physical mechanisms for the effect of alloying elements. As mentioned in connection to the O-C model the elements can contribute to the change of stacking fault energy (SFE) of austenite and make the formation of stacking faults and stacking fault intersections more or less likely. The SFE can be estimated by calculating the driving force for HCP-martensite (−ΔG FCC→HCP RT ) formation using the following expression [67]: where, ρ A , ΔG FCC→HCP , E strain and σ denote the density of atoms on {111} FCC (mol·m −2 ), chemical free energy change from FCC to HCP, the strain energy generated by the formation of HCP from FCC, and HCP/FCC interfacial energy, respectively. If ρ A and σ are assumed constant and E strain assumed negligible [68] we can estimate the SFE change by calculating (−ΔG FCC→HCP RT ), i.e. SFE is increasing with decreasing driving force for HCP-martensite formation.
The other physical mechanism of the effect of alloying elements on SIM is the driving force for the actual BCC-martensite formation, i.e. SIM formation. This can be evaluated by calculating the driving force for BCC-martensite formation.
Hence, both the chemical driving force to form BCC-martensite (−ΔG FCC→BCC RT ) and HCP-martensite (−ΔG FCC→HCP RT ) at room temperature was calculated using Thermo-Calc 2019a [69] with the TCFE9 database [70]. The effects of Ni and Cr content on the calculated driving forces are presented in Fig. 8. A normalized composition of Fe-xNi-yCr-0.1C-0.02 N-1.5Mn-0.5Si-0.3Mo is selected, Ni is varied from 6 to 12 mass percent, and Cr is varied from 16 to 22 mass percent. This host composition is selected to be representative of the 300 alloy grades and then with relevant variations of Ni and Cr. It can be seen that increasing Ni reduces the driving force for BCC-martensite formation significantly; the effect on SFE is lower but also with an increasing tendency according to the thermodynamic calculations. It is known from experimental work that Ni has a strong effect on the austenite stability towards BCC-martensite formation with increasing stability when Ni content is increasing. It has also been shown in prior experimental and ab initio works [51] that Ni has a strong influence on the SFE, which increases with increasing Ni content. That in turn lowers the tendency for shear band formation that is known to aid BCC-martensite formation [51]. The effect of Ni on SFE is not very clear from the thermodynamic calculations here, but looking at the combined effect of Ni on both BCC-martensite and SFE it is reasonable that Ni should have a strong effect on SIM as predicted by the ML model. Cr on the other hand has a much more marginal effect on SIM according to the ML model, and studying the thermodynamic calculations it can be seen that increasing the Cr content decreases the SFE and the driving force for BCC-martensite formation. These two effects would counteract each other and, moreover, the effect of Cr on the driving force for BCC-martensite formation is much lower than the effect of Ni. These predictions support the ML model feature importance predictions, indicating that Cr is much less important than Ni when it comes to austenite stability versus SIM formation. This insight is also supported by, for example, the review on SFE in steels by A. Das [71], where his analysis of the literature data indicated that Ni has a strong positive effect on SFE whereas Cr has a weak, almost negligible, positive effect on SFE.
The effect of the other elements could also be analyzed in a similar way as for Ni and Cr, but as indicated by the analysis of these two major elements, the thermodynamic predictions may not be fully  reliable for minor elements and we therefore refrain from further discussions on this point. It should though be noted that the importance of C is lower than expected. However, the majority (over 97.5%) of the C contents in the dataset ranges from 0.017 to 0.14, and with this narrow range of C contents it is reasonable that its feature importance is not so high. It can also be noted that A. Das [71] predicted that the effect of C on SFE is negligible based on the analysis of the literature data.

Summary
A fully predictive machine learning (ML) model for strain-induced martensite (SIM) formation in austenitic steels has been developed. Chemical composition, temperature and strain are input and output is the SIM fraction. The method applied to develop the predictive model is a combination of physical modelling using the Olson-Cohen model and ML ensemble methods. The model shows promising results for predictions on single-phase and multi-phase steels containing austenite. Further development of the model to include additional features such as strain rate, stress state and austenite grain size will facilitate widespread implementation of the model for steels. The current model can readily be combined with thermodynamics and kinetic calculations to evaluate austenite stability in multiphase steels or used in thermomechanical simulations to predict microstructure and properties after, for example, forming.
CRediT author statement WM: Database establishment, design the work, discussion, writing the manuscript, reviewing and editing. MR: model establishment and testing, discussion; FR: model modification and benchmark; JO: discussion, reviewing and editing; PH: design the work, reviewing, discussion and editing.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.