Data-Driven ANN-Based Predictive Modeling of Mechanical Properties of 5Cr-0.5Mo Steel: Impact of Composition and Service Temperature

Abstract

The mechanical properties of steel are intricately connected to their composition and service temperature. Predicting these properties across different work temperatures using traditional statistical methods, algorithms, and equations is highly challenging due to these complex interdependencies. To address this, we developed an artificial-neural-network (ANN) model to elucidate the relationships between composition, temperature, and mechanical properties of 5Cr-0.5Mo steels. Our model demonstrated high accuracy, with minimal percentage errors in predicting YS, UTS, and El (%)—3.5%, 0.97%, and 1.9%, respectively. The ANN predictions are realistic and closely match the experimental results. We propose an easy-to-use model’s GUI to predict steel composition to achieve desired properties at any temperature. The ANN model’s findings offer valuable insights for researchers and designers, aiding in developing steel components with optimized properties. This technique is expected to significantly enhance the planning of practical experiments and improve material performance overall.

1. Introduction

The drive to improve the thermal efficiency of power-plant components has led to the development of Cr-Mo steels as important construction materials. These steels are highly valued for their excellent mechanical properties at elevated temperatures and their strong resistance to corrosion, making them ideal for use in high-stress, high-heat environments [1]. Such steels are used in components like heater tubes, vessels, and pipes, where temperatures typically range from 300 to 650 °C and are subjected to high stresses over extended periods of around 30 years [2]. The main objective of these steel components is to function reliably throughout their lifespan. An analysis of material failures in the power generation sector indicates that 81% of these failures are attributed to high-temperature exposure, predominantly influenced by mechanical factors. Within this category, 65% of failures are attributed to short-term material failures, such as tensile, compressive, flexural, and shear failures [3].

Conversely, a smaller proportion (~9%) is due to long-term exposures related explicitly to creep phenomena [4]. The mechanical properties of materials are highly dependent on alloying additions and operating temperatures [5,6,7]. Prolonged exposure to high temperatures can precipitate various secondary phases and carbides [8,9,10]. These precipitation and microstructural changes can have a notable impact on the ultimate mechanical properties of the material [11,12]. The grain size of austenite also influences the final mechanical properties of the steels [13]. Furthermore, certain alloying additions have the potential to segregate along grain boundaries, which can significantly influence the mechanical properties of the material [14]. Investigating the effect of each alloying element through experimental methods and assessing its influence on mechanical properties at different temperatures is impractical and would be significantly time-consuming, labor-intensive, and costly. In a similar vein, designing steel compositions that achieve the desired properties over an extended period is also highly challenging [15]. Recent regulations to reduce greenhouse gas emissions in the power sector [16] have made accurate lifespan prediction of these materials even more crucial. Therefore, any new technique or model that can significantly reduce costs by minimizing the number of experimental trials during the design phase is highly welcomed in materials science. Due to the rapid advancements in computational power and intelligent algorithms in recent years, data-driven methods are now commonly used to predict the behavior of structural materials. The significance of such methods lies in their ability to establish correlations among interdependent parameters, like composition and operating temperature, thereby enabling researchers to comprehend these intricate relationships effortlessly [17]. This capability aids in effectively guiding actual experiments, facilitating a more streamlined approach to research and development.

Machine-learning techniques, particularly artificial neural networks (ANNs), have been widely used to predict various alloy systems’ mechanical behavior, as documented in several studies [18,19,20,21,22]. Specifically, ANN models have been applied to forecast the mechanical properties of vessel steels [23,24], austenitic stainless steels [25], maraging steels [26], and cast alloys [27]. The creep behavior of Cr-Mo steel has also been reported by the magnetic Barkhausen emission technique [28]. However, the impact of composition and service temperature on the tensile properties of 5Cr-0.5Mo steel systems requires further elucidation.

This study aims to develop a novel ANN-based predictive model for the mechanical properties (YS, UTS, and El) of 5Cr-0.5Mo steels. The key objectives are as follows:

(i)

Development of an Optimized ANN Model: To accurately predict mechanical properties (YS, UTS, and El) of 5Cr-0.5Mo steels as a function of composition and temperature (room temperature to 700 °C).

(ii)

Input–Output Relationship Analysis: To analyze the relationships between input variables (composition and temperature) and output properties using Pearson correlation and sensitivity analysis, thereby identifying key influential parameters.

(iii)

User-Friendly GUI Development: To develop a graphical user interface (GUI) based on the ANN model, allowing users to predict mechanical properties for any given composition and temperature, supporting efficient material design and experimental planning.

Through the graphical user interface (GUI), users can easily modify the composition to achieve desired properties for a specific service temperature. Alternatively, if users wish to determine the properties of a particular steel composition at a given service temperature, they can simply input the known composition and temperature to predict the corresponding mechanical properties.

2. Materials and Methods

2.1. Data Collection

In the present investigation, data containing the chemical composition, service temperature, and mechanical properties of 5Cr-0.5Mo steel were collected from the National Institute of Materials Science (NIMS), Japan. Various Cr-Mo steel ingots were produced using a basic electric arc furnace, with the compositions provided in Supplementary Materials. Tensile test specimens, each with a diameter of 10 mm and a gauge length of 50 mm, were extracted from the final bar-shaped product. The tensile tests were conducted by following the ISO 9513 standard, using an extensometer at a strain rate of 10⁻³ s⁻¹. The database includes a range of compositions and corresponding mechanical properties tested at temperatures ranging from 25 °C to 700 °C.

The inputs for the current model include (i) the chemical composition (wt.%) of the steel, which encompasses 11 different alloying elements (C, Si, Mn, P, S, Ni, Cr, Mo, Cu, Al, and N) and other non-metallic inclusions (NMI); (ii) austenite grain size (um); and (iii) a series of test temperatures (25 °C, 100 °C, 200 °C, 300 °C, 400 °C, 500 °C, 600 °C, 650 °C, and 700 °C). The model outputs the steel’s tensile properties (YS, UTS, El (%), and RA (%)). The entire experimental dataset consists of 99 samples. Of these, 81 datasets were used to develop the ANN model, while 18 were reserved for testing the model’s accuracy. The details of all the variables in the training and testing datasets, including the minimums, maximums, mean values, and corresponding standard deviations, are presented in

Table 1. Statistical data of input and output variables of Cr-Mo steel used for ANN Modeling.

Variable	Training Data (81 Datasets)				Testing Data (18 Datasets)
	Minimum	Maximum	Mean	Standard Deviation	Minimum	Maximum	Mean	Standard Deviation
Fe	93.35	93.82	93.55	0.19	93.35	93.82	93.55	0.19
C	0.09	0.12	0.108	0.012	0.090	0.120	0.108	0.013
Si	0.27	0.37	0.331	0.035	0.270	0.370	0.311	0.035
Mn	0.44	0.56	0.499	0.048	0.440	0.560	0.499	0.049
P	0.007	0.022	0.015	0.005	0.007	0.022	0.015	0.005
S	0.005	0.012	0.007	0.003	0.005	0.012	0.007	0.003
Ni	0.043	0.083	0.060	0.016	0.043	0.083	0.060	0.016
Cr	4.61	5.02	4.834	0.151	4.610	5.020	4.834	0.154
Mo	0.49	0.52	0.503	0.011	0.490	0.520	0.503	0.011
Cu	0.05	0.13	0.072	0.031	0.050	0.130	0.072	0.031
Al	0.004	0.008	0.006	0.001	0.004	0.008	0.006	0.001
N2	0.0104	0.0178	0.014	0.003	0.0178	0.014	0.014	0.003
AGS (µm)	4.800	6.900	5.967	0.583	6.900	5.967	5.967	0.596
NMI (wt.%)	0.03	0.100	0.086	0.045	0.03	0.100	0.086	0.045
Temp. (°C)	25	700	400	220	100	700	436	198
YS (MPa)	82	425	238	94	87	384	223	93
UTS (MPa)	131	600	369	125	134	494	351	117
EL (%)	21	80	36	13	21	64	37	14
RA (%)	68	96	83	8	75	97	84	8

for both the training and testing datasets.

2.2. Correlation Analysis of Inputs and Outputs

A heatmap in Figure 1 illustrating the linear correlations between inputs and outputs can be created using the Pearson correlation coefficient [29]. The correlation values range from −1 to 1 for simplicity. The colored heatmap allows for quick and easy identification of correlations, with the color intensity of each square representing the strength of the linear relationship between input and output variables. Darker colors indicate a stronger correlation, close to either 1 or −1, while lighter colors suggest a weaker correlation [30]. A positive value indicates a direct relationship between two variables, while a negative value signifies an inverse relationship. A value close to zero, such as 0.001 between Ni and EL (%) in the heatmap, represents a fragile and insignificant relationship.

The correlation coefficient of any parameter with itself is always +1, which appears along the diagonal of the heatmap. While detailing the correlation of each element with others is beyond the scope of this paper, the overall trends can be easily observed through the color variations in the heatmap. Sulfur and phosphorus form non-metallic inclusions [31], indicated by the red color in the heatmap, signifying a strong relationship between these elements and non-metallic inclusions.

2.3. Preprocessing of Data

The experimental dataset consists of 99 samples, as shown in Supplementary. All the variables were normalized to a range between 0.1 and 0.9 to ensure that all input features have consistent scaling, which improves the stability and performance of the model. Normalization was done using the normalization function represented by Equation (1) [32].

$${\mathrm{x}}_{\mathrm{n}}=0.1+0.8{\displaystyle \frac{\mathrm{x}-{\mathrm{x}}_{\mathrm{m}\mathrm{i}\mathrm{n}}}{({\mathrm{x}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\mathrm{x}}_{\mathrm{m}\mathrm{i}\mathrm{n}})}}$$

(1)

Here, the normalized values of x are x_n, and x_min and x_max are the inputs’ minimum and maximum values, respectively.

After achieving the best-trained network, all normalized data are reverted to their original values using Equation (2) [32].

$$\mathrm{x}={\mathrm{x}}_{\mathrm{m}\mathrm{i}\mathrm{n}}+\left({\mathrm{x}}_{\mathrm{n}}-0.1\right)\times ({\mathrm{x}}_{\mathrm{m}\mathrm{a}\mathrm{x}}-{\mathrm{x}}_{\mathrm{m}\mathrm{i}\mathrm{n}})/0.8$$

(2)

2.4. Development of the ANN Model

ANN modeling is inspired by the human brain, where interconnected neurons adjust their responses based on inputs. In an ANN, there are typically three layers: input, hidden, and output layers. This model’s input layer includes composition, non-metallic inclusions, austenite grain size, and temperature. The outputs are YS, UTS, El (%), and RA (%). The hidden layers, positioned between the input and output layers, contain a certain number of nodes. Part of the data is used to train the ANN, enabling it to produce highly accurate outputs from new, unknown inputs. The model is trained by adjusting the network to align the computed outputs with the experimental values. The primary focus during model training is on minimizing the mean squared error (MSE), which represents the average error in output predictions (Etr), calculated by using Equations (3) and (4) [33].

$$\mathrm{MSE}={\displaystyle \frac{1}{\mathrm{P}}}{\sum }_{\mathrm{P}}{\sum }_{\mathrm{i}}{{(\mathrm{T}}_{\mathrm{i}\mathrm{p}}-{\mathrm{O}}_{\mathrm{i}\mathrm{p}})}^2$$

(3)

$${\mathrm{E}}_{\mathrm{tr}}(\mathrm{y})={\displaystyle \frac{1}{\mathrm{N}}}{\sum }_{\mathrm{i}=1}^{\mathrm{N}}\left|{(\mathrm{T}}_{\mathrm{i}}(\mathrm{y})-{\mathrm{O}}_{\mathrm{i}}(\mathrm{y}))\right|$$

(4)

Here, E_tr(y) = Average error in prediction of training and testing sets for output parameter y.

N = Number of datasets, T_i(y) = Targeted output, and O_i(y) = Output calculated.

The training program utilized the C programming language (C99), while Java (1.4) was employed to design the graphical user interface. A comprehensive explanation of ANN-based modeling in materials science is given in another source [34]. Figure 2 depicts the basic structure of the ANN model, which includes input, hidden, and output layers. The number of neurons in the input and output layers matches the number of input and output parameters, respectively. The entire experimental dataset was split into training and testing sets to train the model. The training data were used to identify the optimal model architecture, including the number of hidden layers, neurons per hidden layer, momentum term, learning rate, and number of iterations.

The critical steps in developing the ANN model are as follows:

(i)

Selecting the number of hidden layers and neurons;

(ii)

Determining the momentum term and learning rate;

(iii)

Choosing the number of iterations.

Figure 3 presents a comprehensive flowchart of the investigation using the developed model. It outlines the process of data division into training and testing sets, normalization of values, operation of the ANN within hidden layers, selection of the optimal model, and subsequent accuracy determination and prediction of mechanical properties for unseen data.

2.5. Optimization of the ANN Model

The accuracy and effectiveness of the developed model rely on its training phase, which entails meticulously selecting the optimal configuration of hidden layers and neurons. In the current study, the model was trained using configurations with single and two hidden layers, where the number of neurons in each layer ranged from 2 to 15. The MSE was computed for each configuration of single and two hidden layers, and the results are illustrated in Figure 4. It is evident from the figure that the MSE decreases as the number of hidden neurons increases.

This trend indicates that training the model with two hidden layers leads to a notable reduction in error compared to using a single hidden layer. We obtained minimum testing errors with 2 hidden layers and 12 neurons (MSE of 0.00149 and Etr of 6) (shown in Figure 4). With two hidden layers, we varied the momentum term from 0.05 to 0.95 and observed the minimum error occurred with a 0.95 value (MSE of 0.001136 and Etr of 4.395) (shown in Figure 5a,b). After fixing the momentum term, the learning rate varied from 0.05 to 0.95, and a 0.45 learning rate was found; the minimum error was achieved with an MSE of 0.000698 and Etr of 10.353 (shown in Figure 5c,d). The optimum ANN model was obtained based on the MSE and average error in the prediction of the blind 18 testing datasets. The GUI design of the ANN model was built based on the connective weights of the optimum ANN model. The number of iterations required for ANN training depends on the complexity of relationships between input variables and mechanical properties (21, 22, 23), such as YS, UTS, El (%), and RA (%). Simpler properties like YS and UTS, which correlate strongly with material composition and processing parameters, converge faster due to their relatively straightforward patterns. In contrast, El (%) and RA (%) exhibit more complex dependencies on microstructural factors and secondary phenomena, necessitating additional iterations for accurate predictions. The variability in iteration requirements reflects the ANN’s ability to handle diverse levels of complexity effectively.

2.6. Sigmoid Output Transfer Function

In our developed ANN model, we implemented the sigmoid function as the output transfer function [35]. This choice of activation function enhanced the predictive accuracy of the trained model configured with a 2-12-12 architecture. The transfer function is a crucial element in neural networks, providing the capability to manage non-linear relationships. The complexity and performance of neural networks are heavily influenced by selecting a suitable transfer function [36]. The initial ANN model synaptic weights are randomly generated between −0.5 and +0.5. Figure 6 illustrates the transformation of weights with the number of iterations. The optimum model weight values varied between −16 and +16.

We varied the number of iterations to find the testing error deviations. We observed 10,650, 7350, 6100, and 1050 iterations for YS, UTS, El (%), and RA (%), respectively. As the number of iterations increases to 10,650, there is a significant reduction in error values, and the curve gradually adopts a sigmoid-function shape. This transformation is accompanied by an improved distribution of weights, demonstrating the effectiveness of the sigmoid activation function in modeling the provided database.

3. Results and Discussion

3.1. Prediction of Mechanical Properties Using ANN

After training the network with 81 datasets, the remaining 18 datasets are used to test the model performance. Figure 7a,c show the comparison among actual or experimental mechanical properties (YS and El) of training datasets. Prediction of mechanical properties of 5Cr-0.5Mo steels for specified input parameters (composition, test temperature) was achieved. Pearson’s r and adj. R-squared values are nearer to one for training datasets of all properties. A similar comparison for the testing dataset is presented in Figure 7b,d. The predictions with unseen data are very accurate for the YS. The greater scatter in the El (%) resulting from the testing dataset can be attributed to the intrinsic variability of the elongation property across different steel compositions and testing conditions.

The MSE and average error values were calculated based on the differences between predicted and experimental mechanical properties, expressed in their respective dimensional parameters, as shown in Figure 8a–d. For YS, the percentage average error in prediction is 4.655%, while for El (%), it is 3.76%. These errors indicate the model’s ability to achieve high accuracy in predicting tensile properties, with most predictions exhibiting an average error of less than 5%, as shown in Figure 8a,b for YS and Figure 8c,d for El (%). The ANN model demonstrates exceptional accuracy in modeling and optimizing the relationships between complex parameters, making it highly effective for improving new alloy systems with desired properties.

Mechanical properties were predicted as a function of temperature using ANN model predictions. Results from the model show that as the temperature increases from 0 °C to 700 °C, the two strength properties (YS, UTS) gradually decrease irrespective of the steel composition, as shown in Figure 9a,b. From the metallurgical point of view, the higher dislocation motion with the increase in temperature is the main factor that influences the plastic behavior of steel [37]. An increase in the working temperature of steel leads to quick movement in dislocations, and as a result, steel deforms more plastically. So, a temperature rise will reduce the strength properties (YS, UTS). Moreover, at around 400 °C, there is a slight increase in YS and UTS, which can be attributed to dynamic recovery or carbide/precipitation hardening mechanisms at this temperature, which may have enhanced the strength and delayed the onset of softening temporarily [38]. On the other hand, Figure 9c,d show El (%) and RA (%) with the function of temperatures. It was observed that both properties are increased linearly at elevated temperatures of higher than 400 °C. This is due to more slip planes becoming active at higher temperatures, thereby increasing the material’s ductility [39]. From the outcomes of ANN, it is clear that model predictions are matches with experimental results. The available data are with 25, 100, 200, 300, 400, 450, 500, 550, 600, 650, and 700 °C only. However, the model can predict at any temperature with small errors.

3.2. Single Variable Sensitivity Analysis

The ANN model provides the sensitivity analysis by varying any of our chosen independent variables on output parameters. As here, we can get several results from the model, we consider the effect of temperature and carbon on UTS and El (%).

3.2.1. Effect of Temperature

The mechanical properties of 5Cr-0.5Mo steel are highly temperature-dependent. At higher temperatures, yield and tensile strength decrease due to the role of temperature in the development and dissolution of precipitates [40]. At around 400 °C, fibrous M₂C carbides disappear, and massive grain boundary M₂₃C₆ carbides form, which can develop at grain boundaries or within grain interiors [41]. The sequence of carbide precipitation is ε-carbide → cementite → M₂C → M₇C₃ → M₂₃C₆ → M₆C [42]. These M₂₃C₆ carbides require a large number of Mo atoms, leading to the dissolution of carbides into the ferrite matrix after 400 °C [43]. Depending on the temperature and exposure duration, these carbides can vary in shape, being either spherical or elongated, and can exhibit varying compositions, e.g., Cr-rich and/or Mo-rich [44]. The UTS values initially decreased slowly, then showed a steep decline after 400 °C due to carbide dissolution. Similarly, El (%) initially decreased until 400 °C, then increased. The graphs in Figure 10a,b display the experimental and predicted values and trends of UTS and El (%) against different temperatures, proving the model’s intelligence without explicit metallurgical inputs to the ANN model.

3.2.2. Effect of Carbon

Carbon plays an important role in strengthening Cr-Mo steels through precipitation strengthening by forming carbides, such as Fe₃C, M₇C₃, M₂₃C_6, and Mo₂C [45]. However, it is worth mentioning that these carbides form at higher temperatures, and prolonged heating can lead to coarsening, resulting in decreased strength [8]. The graphs generated by our developed ANN model demonstrate the effect of carbon content on UTS and El (%). The ANN model accurately predicted the trends for both UTS and elongation at a randomly selected temperature of 700 °C, as shown in Figure 10c,d. It is well known that alloys with higher UTS typically exhibit lower ductility [7]. Consequently, while an increase in carbon content enhances strength, it also results in a decrease in elongation, a relationship accurately predicted by our model.

3.3. Two Variable Sensitivity

The combined effect of carbon and silicon on the mechanical properties of Cr-Mo steels is significant. Carbon, crucial for precipitation strengthening, forms various carbides that enhance YS and UTS. An increase in carbon content can result in a higher volume fraction of carbides, leading to improved YS and UTS [46]. Most of the carbides, like cementite, are brittle and can reduce ductility [47]. Silicon acts as a deoxidizer and contributes to strength through solid solution strengthening but can also reduce ductility at high concentrations [48]. The contour graphs in Figure 11a–d are generated by our ANN model and illustrate the intricate interplay between C and Si. These graphs plot C and Si against YS, UTS, El (%), and RA (%), revealing clear trends. Increased levels of C and Si lead to higher YS and UTS, confirming their strengthening effects. However, these increases come at the expense of El (%) and RA (%), indicating a trade-off between strength and ductility. Most importantly, the ANN model was not fed the explicit relationship between inputs and outputs. Nevertheless, the well-trained model is capable of capturing and mapping of these relationships.

3.4. Graphical User Interface (GUI)

The developed ANN model has been integrated into a user-friendly GUI, featuring fourteen input parameters and four output parameters (predicted YS, UTS, EL, and RA). Users can manually adjust the input parameters and calculate all four outputs with a single button click. For linear visualization, there is a button to select the desired seed and another to save the plot. Users can save the plot data to a new file, with the number of points customizable. Surface plots, which show the combined effect of multiple inputs, can also be generated similarly for easy visualization and understanding. Additionally, bar plots displaying both input and output parameters are accessible on the same screen with a single click. Input values can be adjusted by clicking and dragging the respective bars. This makes the model highly intuitive and easy to use. We have included one screenshot of the GUI of our developed model, as shown in Figure 12. Figure 12 provides a detailed visualization of how mechanical properties change with temperature. The calculations allow users to observe subtle variations and trends by displaying temperatures at finer intervals. The predicted mechanical properties are presented on a normalized scale, making comparing and analyzing the data across different conditions easier. This feature enhances the usability and interpretability of the model, facilitating more precise and informed decision-making in materials engineering.

4. Conclusions

A neural network model has been developed to estimate the mechanical properties of 5Cr-0.5Mo steel based on its chemical composition (C, Si, Mn, P, S, Ni, Cr, Mo, Cu, Al, and N), the presence of non-metallic inclusions, austenite grain size, and test temperatures (ranging from 25 to 700 °C). The model’s predictions align well with experimental data, accurately reproducing numerous observed trends by capturing complex nonlinear relationships. The model accurately predicted the mechanical properties, with a Pearson’s r of 0.99853 and an adj. R² of 0.99763 for the YS in the training dataset, a Pearson’s r of 0.98479, and an adj. R² of 0.99281 for the testing dataset. Similarly, for El (%), the model achieved a Pearson’s r of 0.99265 and an adj. R² of 0.98517 for the training dataset, a Pearson’s r of 0.93112, and an adj. R² of 0.85868 for the testing dataset, demonstrating the model’s efficiency. The percentage errors in prediction for YS, UTS, and El (%) are 3.5%, 0.97%, and 1.9%, respectively. The contour plots generated by the model allow for visual optimization of YS, UTS, El (%), and RA (%) of the steel under various conditions. The ANN-based GUI model is a practical tool for the steel industry, enabling users to predict key mechanical properties (YS, UTS, and El) of 5Cr-0.5Mo steels based on composition and temperature. Its user-friendly interface supports alloy design, process optimization, and material selection, allowing engineers and researchers to make informed decisions without requiring expertise in machine learning.