Making reliable predictions of the mechanical behavior of alloys with a prolonged service life is beneficial for many structural applications. In this work, we propose an interpretable machine learning (ML) approach to predict fatigue life cycles (Nf) and creep rupture life (tr) in titanium-based alloys. Chemical compositions, experimental parameters, and alloy processing conditions are employed as descriptors for the development of gradient boost regression models for log-scaled Nf and tr. The models are trained on an extensive experimental dataset, predicting log-scaled Nf and tr with a very small root mean squared error of 0.17 and 0.15, respectively. An intuitive interpretation of the ML models is carried out via SHapley Additive exPlanations (SHAP) to understand the complex interplay of various features with Nf and tr. The SHAP interpretation of the ML models reveals close agreement with the general creep equation and Wöhler curve of fatigue. The approach proposed in this study can accelerate the design of novel Ti-based alloys with desired properties.
Material failure is the loss of the load carrying capacity of a structure. The analysis of material failure is essential for determining the integrity of a material during its applications. The long-term load and the extreme temperature environment reduce the service life of a material. Materials operating at high temperatures and under cyclic loads could potentially be subjected to significant creep and fatigue degradation. Therefore, creep and fatigue failures have become the main factors that threaten the integrity of materials. The experimental approaches to determine creep and fatigue deformation in materials are highly time and resource intensive.1–4 Furthermore, intuitive experimental approaches and traditional life prediction models cannot fully consider the complex relationship between all parameters and the corresponding properties. To ameliorate the limitations of traditional approaches, data science has emerged as a powerful tool for making reliable estimates at a faster scale. Machine learning (ML) enables the research community to understand the future trends by learning from the past existing data. Numerous achievements have been reported involving the implementation of ML in materials science, for example, property prediction,5–11 development of ML potentials,12 and discovery of new materials.13 Various reports demonstrated the implementation of ML models in the prediction of failure analysis in structural materials.14–18 Most ML-based studies focus mainly on predicting the lifetime of failure of a specific composition or class of materials. However, there are very few reports that address the effect of independent compositions in predicting the failure lifetime in alloys. In recent times, researchers have focused on the search for alloys with high toughness and structural integrity, particularly with regard to dynamic load and extreme operating conditions. Titanium alloys have proven to be a potential candidate with widespread applications in various fields due to their light weight, high strength, good corrosion and heat resistance, and high strength-to-weight ratio.19 Because titanium alloys are the workhorse of the aerospace and aviation sectors, their service life and failure properties should be investigated. Numerous studies have been carried out to understand the effect of various parameters and microstructures on the mechanical and failure properties of titanium alloys.20–25 However, there are no reports incorporating the data-driven prediction of fatigue and creep lifetime in different classes of titanium alloys.
In this work, we propose two independent and interpretable ML models for an accelerated prediction of fatigue life cycles (Nf) and creep rupture life (tr) in titanium alloys. Our approach focuses on the prediction of Nf and tr for different classes of titanium alloys without any constraint on the alloy composition. The database is comprised of experimental data collected for 399 titanium-based alloys with alloy composition, heating conditions, and experimental parameters as descriptors. The Pearson correlation coefficient (PCC) is implemented to select the relevant features. The Gradient Boost Regression (GBR) model is trained for the prediction of log-scaled Nf and tr. The ML predictions are highly accurate, with a small root mean squared error for both target properties. Furthermore, a SHapley Additive exPlanations (SHAP) interpretation of the ML models shows close agreement with empirical materials science and physics-based models. The developed ML model serves as an interpretable model for different titanium alloy compositions. The approach emphasizes the correlation of individual features with the target properties, unraveling scientific insights for the faster prediction and design of alloys with the desired service life.
The training of the ML models for predicting log-scaled Nf and tr is carried out using an ensemble-based Gradient Boost Regression (GBR) model.26 The performance of the model is estimated by calculating statistical regression metrics such as the root mean squared error (rmse) and the coefficient of determination (R2). The model is trained by performing shuffle-split cross-validation (CV) with a 90:10 train/test split ratio over 3000 random trials. The hyperparameters are optimized using the grid search method. The best performing model is selected based on the lowest train/test rmse and the highest R2 in all random trials. The interpretability of the ML models is explained using the SHapley Additive exPlanations (SHAP) method to identify the influence of each descriptor on its target properties.27 The schematic workflow for developing ML models and incorporating SHAP-based insights is presented in Fig. 1. All the codes related to ML are built using scikit-learn (version 0.23.2) and PyCaret (version 2.3.10) packages of the Python (version 3.9) programming language. The SHAP (version 0.40.0) package is utilized for interpreting the ML models. Additional details of the methods are added in Sec. 1 of the supplementary material file.
III. RESULTS AND DISCUSSION
A database is developed by the collection of data from the experimental literature, having fatigue life cycles and creep rupture life in the range of to 108 cycles and ∼0.01–1744 h, respectively, at different temperatures (details mentioned in the supplementary material). The initial database is comprised of a total of 399 titanium-based alloys, with Ti64, Ti6242, and IMI-834 forming the major class. The database is organized into 222 fatigue life data points and 177 creep rupture data points. The feature space for fatigue life and creep rupture covers essential descriptors, including chemical compositions, heat treatment conditions, and experimental parameters (as shown in Tables S1 and S4). The distribution of data points in the database is plotted as shown in Figs. S1 and S5 for Nf and tr, respectively. The data points located far from the interquartile region represent outliers. The Z score outlier detection method is implemented to remove these data points to minimize misleading representation of data. Standardization is performed to nullify the large variation in the values of different features.
A. Prediction of fatigue life
The first attempt is to develop a ML model for the prediction of fatigue life cycles (Nf) of Ti-based alloys. The descriptors used for the prediction of Nf are summarized in Table S1. The database after the removal of outliers is reduced to a final dataset (X, Y) with 186 uniformly sorted data points, where X = , with n = 22 descriptors, and Y = Nf is the target property. The GBR model outperforms compared to different ML models for the prediction of Nf (as shown in Table S2). Initially, the GBR model is trained, taking all 22 features to consider their influence on log-scaled Nf. This results in a train/test R2 and rmse of 0.99/0.99 and 0.15/0.17, respectively. The selection of the relevant descriptors is important to reduce the complexity and improve the interpretability of the models. Feature selection is carried out by computing Pearson’s correlation of the descriptors with each other. The Pearson correlation coefficient (PCC) between any two descriptors is given by PCC = cov(xi, xj)/σiσj, where cov and σ represent the covariance and standard deviation of xi/j, respectively. The value of |PCC| ranges between 0 and 1, where a higher value indicates redundancy of features. Therefore, one of the two given features with |PCC| > 0.8 is eliminated to remove the linear dependence of features. Next, different combinations of subsets of these 13 PCC selected features are carried out to select the best subset of minimum features required to predict log-scaled Nf of titanium alloys. However, the GBR model shows a higher accuracy with 12 descriptors giving a higher train/test R2 of 0.99/0.99 and the lowest rmse of 0.15/0.17 as shown in Fig. 2(a). The hyperparameters for the GBR model are selected using the grid search algorithm during the learning process, as detailed in Table I. The predictive performance of the model is harnessed using the residual plot as shown in Fig. 2(b). Residuals are defined as the difference between Nf (experimental) and Nf (predicted) values, which is less than 0.4. The |PCC| values of the final 12 features are less than 0.80 and are shown in Fig. 2(c). Next, the learning of the trained model is checked by plotting the learning curves for different train/test splits as shown in Fig. S3. The convergence of R2 and rmse with the increase in train data implies no overfitting of the developed model.
|Model parameters .||Range .||Optimal value .|
|subsample space||[0.0, 1.0]||1.0|
|Model parameters .||Range .||Optimal value .|
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Next, the SHAP feature importance method is adopted to demonstrate the efficacy of our proposed model and find the important features governing the failure mechanisms in titanium alloys. The SHAP summary plot is plotted to unravel the interpretability of the developed model as shown in Fig. 2(d). It can be seen that Al shows a positive and higher importance score, followed by σf, Tsol, Tf, f, ɛtot, H, R, Mo, Tage, tage, and tsol. The high and positive correlation of Al indicates that Nf increases with the increasing concentration of Al. As the dataset primarily spans Ti64, Ti6242, and IMI-834 (α + β) alloy families, which comprise Al as the major element, hence, it has a high impact on Nf prediction. Additionally, Al is an α stabilizer, which dissolves in the matrix Ti, forming solid solution strengtheners as well as precipitate strengtheners, thereby increasing the fatigue strength and cycles to failures.28 Next, Mo and V are primarily present throughout the database, but only Mo appears in the final feature set. The reason is that the PCC correlation between Mo and V is greater than the threshold, making one of them redundant. However, Mo or V can be considered equivalent for the prediction of fatigue life, depending on the composition of the titanium alloy. Considering Mo or V in the development of GBR for the prediction of Nf results in the same accuracy. Apart from the compositional dependence, various other features also show a good correlation with Nf. Fatigue stress (σf) shows a negative correlation with Nf. The variation of σf with Nf highly resembles the commonly characterized S–N curve or the Wöhler curve.29 The S–N curve shows the variation of the stress amplitude with the number of load cycles. The increase in the stress amplitude induces an additional and progressive plastic deformation in the material, leading to its failure. The high stress amplitudes usually cause a low cycle fatigue (LCF) failure, while the decrease in the stress amplitude causes a high cycle fatigue (HCF) failure. In 2006, Tokaji showed that the fatigue strength decreased at elevated temperatures compared to ambient temperature in Ti6Al4V alloy.30 It is also observed that for LCF (usually 105 cycles), the fatigue strength decreases with the increasing temperature, while for HCF (105 cycles), it remains nearly the same for both temperatures. The reason is that at low cycles with an increased temperature, the crack initiation resistance is less, which produces fatigue damage easily. Figure 2(d) also shows a negative correlation of temperature with fatigue life as reported. Next, for the effect of strain on Nf, our results match well with the literature, where it is observed that high strain accumulation rates increase plastic deformation, thus leading to a significant reduction in fatigue life.31 The loading frequency (f) is another important feature that influences Nf and shows that Nf increases with an increase in f, which is similar to the results observed in the literature.32 As reported for a given stress amplitude at lower frequencies, dislocations have more time to overcome obstacles through thermal activation, increasing local plastic strain accumulation and leading to a shorter fatigue life.33 Higher frequencies will require higher loads to produce the same strain range. Since fatigue is a plasticity-induced phenomenon, this could explain the increase in fatigue strength at higher frequencies.34
The fatigue limit is highly affected by the stress ratio (R). It is observed that as the stress ratio decreases, the stress amplitude increases and, hence, implies a greater crack growth rate and fatigue damage, which is also captured by our model [Fig. 2(d)].35,36 Also, it has been reported that the crack growth rate of short cracks is faster than that of long cracks.33,37 The heat treatment (solution treatment and aging) determines the formation of precipitates and microstructural changes. The formation of precipitates during heating conditions resists the fatigue crack growth, thereby increasing the number of cycles to failure. The SHAP summary plot in Fig. 2(d) shows that Al, σf, and Tf are important features for the prediction of Nf in titanium alloys. Therefore, the analysis of local interpretation of these features with Nf is carried out in Fig. 3. Both LCF and HCF phenomena are analyzed under ambient and elevated temperatures. In LCF [as shown in Figs. 3(a) and 3(b)], σf and Tf vary in the opposite manner for both conditions. At ambient temperature, a higher stress value reduces Nf (pushes the base value toward the left or lower side). In contrast, at elevated temperatures, a lower stress value can lead to higher Nf (pushes the base value toward the right or higher side). The base value is the mean value of log-scaled Nf. Furthermore, a similar trend is observed for the case of HCF at ambient and elevated temperatures. Additionally, for LCF, Al < 6% is beneficial, and for HCF, the concentration of Al > 6 (in wt. %) is beneficial, as shown in Figs. S2 and S4. A low Al concentration pushes the base value toward lower Nf, leading to LCF, and a high Al concentration pushes the base value toward higher Nf causing HCF. Moreover, the SHAP dependence plots for all descriptors are shown in Fig. S4, depicting the correlation of individual descriptors with log-scaled Nf. The validation of the GBR model is carried out on new and unseen data within the domain of titanium alloys. The validation model shows a good prediction on unseen data with low residuals as shown in Fig. 4 (also shown in Table S3).
B. Prediction of creep rupture life
For the prediction of creep rupture life (tr) in titanium alloys, a similar approach is followed as discussed above. The GBR model gives the highest predictive performance compared to different ML models (as shown in Table S5). Therefore, a GBR model is trained on 130 uniformly sorted data points filtered after eliminating outliers. The GBR model is first trained on all 23 descriptors, giving an R2 of 0.99/0.98 and rmse of 0.13/0.15. Feature selection using the PCC is implemented to reduce the model complexity and to select the most relevant descriptors to predict tr. The features having |PCC| > 0.8 are eliminated, ensuring non-redundancy of descriptors. Next, different subsets of the PCC selected features are carried out to find the best subset of minimum features required to predict tr. The reduced set of 12 descriptors is used to train the GBR model, which gives a higher train/test R2 of 0.99/0.98 and lower rmse of 0.14/0.15 as shown in Fig. 5(a). The optimized hyperparameters for the model are listed in Table II. The residuals for log-scaled tr (predicted) and tr (experimental) are observed to be less than 0.25, indicating a good prediction accuracy, as shown in Fig. 5(b). The PCC coefficients for the top 12 descriptors are shown in Fig. 5(c), ensuring that they are not linearly correlated. The accuracy of the model and the fitting trend are shown using the learning curves as shown in Fig. S7. The convergence of R2 and rmse with the increase in train data implies no over-fitting of the model.
|Model parameters .||Range .||Optimal value .|
|subsample space||[0.0, 1.0]||1.0|
|Model parameters .||Range .||Optimal value .|
|subsample space||[0.0, 1.0]||1.0|
Furthermore, SHAP interpretations are carried out to correlate the highly accurate ML predictions with the materials science and physics of titanium alloys. The SHAP summary plot [shown in Fig. 5(d)] shows the correlation of the features with tr. It is observed that and Tcreep descriptors have the highest importance score and a negative correlation with tr. This implies that an increase in and Tcreep reduces the tr in Ti alloys. Although Mo and V are selected in the final features set, they are not consistently present in the database. Mo is present in Ti6242, IMI834, and other Ti-based alloys, but it is absent in Ti64 alloys, where V is present. Therefore, Mo and V are considered equivalent representatives and can be used depending on the composition of the titanium alloy chosen for the prediction of tr. To understand the role of individual features affecting tr, different stages of creep life curve are discussed. Titanium alloys exhibit a normal creep curve consisting of primary, secondary, and tertiary stages. The experimental studies show that the initial stage or primary creep is related to dislocation generation and strain hardening, which depend on temperature.38 However, the creep deformation mainly occurs during the steady state stage or secondary creep stage. This implies that temperature, stress, and creep rate are relevant parameters influencing the creep rupture in titanium alloys. Generally, the steady-state stage is described using a power law for stress dependence and an Arrhenius law for temperature dependence as defined in the equation
where is the steady state creep rate (1/s), A is a constant related to microstructural features, Q is the activation energy for secondary creep (kJ/mol), σ is the stress (MPa), n is the stress exponent, R is the gas constant (kJ/mol), and T is the temperature (K). The relationship between the steady state creep rate and time to rupture (tr) are given by the Monkman–Grant relationship defined as39
where is the steady state creep rate, tr is the creep rupture life, and M is the Monkman–Grant constant. This relationship shows that the creep resistance decreases with the increase in the strain rate, which is also captured by our model as shown in Fig. 5(d). As known, creep is a temperature-driven phenomenon, showing the tendency to reduce the material life with the increasing temperature. Also, from Fig. 5(d), Tcreep shows a negative correlation with tr, which is in agreement with the report showing that at low stresses and temperatures, the creep rates are lower, resulting in a better creep resistance.38 The creep strain is another important design factor in analyzing the rupture lifetime in titanium alloys. The creep strain occurs in the primary, secondary, and tertiary stages of the creep curve. However, the strain to rupture (ɛr) is observed in the tertiary stage of creep to determine tr. From Fig. 5(d), it is observed that an increase in ɛr increases the creep resistance and, hence, the creep rupture life. The correlation shown in Fig. 5(d) is consistent with the experimental reports that address the decrease in the applied stress, which increases the strain to fracture, thereby increasing the rupture life.40,41 Next, the solution treated and aging conditions are important factors leading to phase transformations, the formation of precipitates, and the change in microstructures.42,43 The Tsol applied above or below the β transus causes the microstructural changes affecting the properties. However, in this study, different titanium alloys have different β transition temperatures; therefore, the absolute correlation of Tsol with the creep rupture life could not be established. For Ti64 alloys, the decrease in the solution temperature increases the primary α volume fraction and decreases the creep rupture life, which is captured by our model.44
Additionally, it is important to emphasize that the trace element concentration is also known to influence the creep behavior in titanium alloys. Elements such as Sn, Mo, and Nb are deliberately added to titanium to improve its creep resistance. These elements are present as β-stabilizer, leading to the formation of precipitates and block the dislocation motion, thereby increasing the creep life. In contrast, C, N, H, and O, when present in small quantities (in the order of ppm), also affect the creep life.45 Iron is known to degrade the creep resistance in titanium alloys. An increase in the Fe content increases the diffusivity, leading to the deterioration of creep properties.46 Therefore, a negative correlation is observed for Fe with tr in Fig. 5(d). The dependence plots of individual descriptors are shown in Figs. S6 and S8, indicating their correlations with tr. Therefore, the developed model captures the essential correlations controlling the creep rupture life in titanium alloys. The model is validated on new and unseen data within the domain of titanium alloys to check its generalizability on new data. The model acquires a good prediction on unseen data points with low residuals, as shown in Fig. 6 (also in Table S6). The prediction and interpretation of the ML models allows us to unravel physical insights by analyzing the past data. Our developed model could capture the empirical creep laws and equations, considering simple compositional and experimental features of titanium alloys. Although the established dataset predicts the creep and fatigue life with a very high accuracy, it certainly does not represent the complexity of microstructures and processing parameters. This can be considered an open area of research to explore the complex microstructures and processing conditions of titanium alloys. Furthermore, extensive studies can be conducted to extract the significance of microstructures and processing parameters on failure life cycles in titanium alloys.
In summary, two independent ML models are developed for the prediction of fatigue life cycles (Nf) and creep rupture life (tr) in titanium alloys. The GBR models are trained on 12 simple compositional and experimental descriptors selected using Pearson’s correlation coefficient. For the prediction of Nf, the GBR model gives a higher train/test R2 and lower rmse of 0.99/0.99 and 0.15/0.17, respectively. Similarly, for the prediction of tr, the GBR model shows an excellent accuracy with a train/test rmse of 0.14/0.15 and R2 of 0.99/0.98, capturing the large variability in log-scaled tr. The SHAP interpretation presents close agreement with the materials science and physics-based models. Using our ML-based approach, we could establish the performance of the S–N curve for fatigue and the power law for creep for different titanium alloys. Additionally, the ML models trained on experimental data can serve as a guide to experimentalists directing the design of titanium alloys with a higher service life. Our approach can also accelerate the prediction of desired properties in other classes of alloys and materials.
See the supplementary material for database information and descriptors used and a summary of the methods used in the study.
The authors acknowledge the support from the Institute of Eminence (IoE) scheme of the Ministry of Human Resource Development (MHRD), Government of India. The authors also thank the support from Materials Informatics Initiative of IISc (MI3). The authors are thankful for the financial support from Aeronautics R&D Board (ARDB), Ministry of Defence (R&D ORGN.), Government of India, under Grant No. ARDB/GTMAP/01/2031993/M/I. The authors thank Materials Research Centre (MRC), Supercomputer Education and Research Centre (SERC), and Solid State and Structural Chemistry Unit (SSCU) at Indian Institute of Science, Bangalore, for providing the required computational facilities. S.S. acknowledges the support from DST, India, through INSPIRE Fellowship (Grant No. IF180007).
Conflict of Interest
The authors have no conflicts to disclose.
Sucheta Swetlana: Formal analysis (lead); Investigation (lead); Writing – original draft (equal); Writing – review & editing (equal). Ashish Rout: Formal analysis (supporting); Investigation (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). Abhishek Kumar Singh: Conceptualization (lead); Supervision (lead); Writing – original draft (equal); Writing – review & editing (equal).