Creep Properties Prediction of Thermal-Exposed CMSX-4 Nickel-Based Superalloy Using Convolution Neural Network and 2-Point Spatial Correlation Analysis

Abstract

In this study, CNN models were developed to predict the changes in creep properties of long-term aged CMSX-4 alloy based on heat treatment time by training deep neural networks with microstructure images of the material. To predict the creep rupture time and fracture strain of specimens heat-treated for 0 to 10,000 hours, the CNN models were trained using BSE images of the specimens and their two-point spatial correlation images. As the heat treatment time of CMSX-4 alloy increases, topological inversion occurs, where the arrangement of the γ phase and γ′ phase changes, leading to significant microstructural changes. When the CNN models, built to predict the creep properties based on microstructural evolution, were trained with 8-bit grayscale BSE raw images, γ-γ correlations, or γ-γ′ correlations, the model trained on γ-γ′ correlations exhibited the best performance in predicting creep rupture time and strain. With the development of CNN models and computational resources, it has become possible to directly learn from raw microstructure images. However, it remains essential to capture microstructures from areas large enough to adequately represent the characteristics of the specimen. In microstructures composed of γ and γ′ phases, two-point spatial correlation analysis serves as a microstructure descriptor, providing sufficient information for artificial neural networks to predict material properties. This study demonstrates such findings and is expected to contribute to various artificial neural network research utilizing microstructure images.

1. Introduction

CMSX-4 is a Ni-based single crystal superalloy widely used as a material for power generation and aviation gas turbines due to its excellent mechanical properties and oxidation resistance at high temperatures and pressures. The precipitation strengthening mechanism by γ′ phase precipitated within the γ matrix phase suppresses deformation at high temperatures and improves the alloy’s strength and creep resistance. However, when exposed to high temperatures for extended periods, even without applied stress, a topological inversion phenomenon occurs where the spatial arrangement of γ and γ′ phases reverses, making the γ′ phase a continuous matrix and the γ phase discontinuous, which adversely affects creep resistance. Various studies have been conducted to analyze the relationship between microstructure and mechanical properties from a Process-Structure-Property-Performance perspective, as the γ/γ′ topological inversion phenomenon involves microstructural changes due to interfacial energy changes and elemental diffusion between the two phases [1–7].

Recent rapid advancements in artificial neural networks (ANN) applied to fields such as image recognition, video processing, and natural language processing have made it possible to develop models that directly connect material microstructures with their properties. Traditional research has focused on deriving relationships that can theoretically explain physical phenomena, either by assuming microstructural evolution mechanisms through methods like phase-field modeling and comparing simulated microstructures with experimentally observed ones, or by extracting features such as grain size and orientation relationships from observed microstructures and linking them to material properties [3, 8–11]. In contrast, data analysis models like convolutional neural networks (CNN) can directly learn from two-dimensional or higher-dimensional images, preserving local information while automatically extracting image features and optimizing parameters. Various types of data can be used for training CNNs, and research has been reported using diverse image data such as actual specimen images measured by optical microscopy or virtual microstructures generated through modeling [11–17]. CNN architectures have also been explored in different ways, including building custom neural networks tailored to material properties or applying transfer learning to fine-tune publicly available large-scale models trained on massive datasets [17–20].

In this study, we predicted creep properties according to high-temperature exposure time by training a CNN with microstructural images of heat-exposed CMSX-4 alloy. Although CMSX-4 is a single crystal superalloy without grain boundaries, the segregation of solute elements during solidification results in different microstructural patterns between the dendritic core (DC) and inter-dendritic (ID) regions. Therefore, a strategy is needed to effectively train the model on heterogeneous microstructural characteristics appearing within the same specimen. While CNN filters (kernels) extract local information within input images, they do not connect local information between different input data. This could be addressed by using large-area images that include both DC and ID regions for training, but this approach is inefficient from both microstructure acquisition and CNN training perspectives. Therefore, this study employed 2-point spatial correlation analysis as a filter to extract local features from the acquired microstructural images. N-point spatial correlation analysis is a method used in statistics and data analysis to analyze how values at specific locations are related to values at surrounding locations, and it is used in various fields including materials engineering, astronomy, and geography [21–24]. We performed 2-point spatial correlation analysis to measure the two-dimensional correlation of γ/γ′ phase distribution in DC and ID regions according to distance and direction, and built a model to predict creep rupture life and strain based on high-temperature exposure time by training a CNN with this data.

2. Experimental Procedure

To obtain creep properties and microstructural images of heat-exposed CMSX-4 alloy, single crystal CMSX-4 specimens with a diameter of 13 mm and length of 170 mm, having the composition shown in Table 1, were manufactured using directional solidification (ALD Vacuum Technologies, ISP 0.5DS/SC/LMC). Specimens with less than 7 degrees misorientation between the single crystal growth direction and the [001] orientation were selected through electron backscatter diffraction (EBSD) analysis for use in creep testing after high-temperature exposure treatment. After standard heat treatment of CMSX-4 alloy (solution treatment and two-stage aging), the specimens were vacuum-sealed in quartz tubes along with pure titanium chips and subjected to high-temperature exposure at 1050°C for 1000, 2000, 5000, and 10000 hours. To investigate changes in mechanical properties with high-temperature exposure time, creep tests were conducted at 1050°C under a load of 160 MPa. Using a field emission scanning electron microscope (JEOL, JSM-7900F), 21 back scattered electron (BSE) images at the same magnification were acquired for each region (DC, ID) per specimen (high-temperature exposure time). The acquired raw images and subsequent 2-point spatial correlation analysis results were used to train a CNN to predict creep rupture life and strain.

Table 1 Composition of CMSX-4 specimen in weight percent (mass%).

3. Results and Discussions

3.1 Creep properties and microstructure analyses

As shown in the creep property changes with high-temperature exposure time in Fig. 1, creep life decreased as exposure time increased, while creep rupture strain increased up to 5000 hours of exposure and then decreased at 10000 hours of exposure (Table 2). The coarsening and topological inversion phenomena of γ and γ′ phases in high-temperature exposed specimens are related to creep life [7]. Figure 2 shows the calculation results of equilibrium phase fraction changes with temperature using the TCNI6 database of Thermo-Calc [25], and Table 3 shows the fraction and composition of each phase at 1050°C.

Fig. 1

Creep test results of heat-treated CMSX-4 alloys.

Table 2 Creep life data of CMSX-4 with high temperature exposure time.

Fig. 2

Equilibrium phase fractions of CMSX-4 alloy calculated with Thermo-Calc with TCNI6 database.

Table 3 Equilibrium phase fractions at 1050°C and their compositions (mass%) calculated by Thermo-Calc with TCNI6 database.

Typically, creep properties of materials are described by expressing relationships between creep stress, steady-state creep strain, and total creep strain as related functions. In contrast, this study trained an artificial neural network using microstructural images of fractured specimens to predict creep life and rupture strain. As shown in Table 3, heat-exposed CMSX-4 material exhibits distinct compositional differences between γ and γ′ phases, with different microstructural patterns in dendritic core and inter-dendritic regions. The γ and γ′ phases have FCC (face-centered cubic) and L1₂ (ordered FCC) structures respectively, with nearly identical lattice parameters, making it impossible to distinguish the two phases through EBSD analysis. However, due to the significant difference in mass per unit volume between the two phases as shown in Table 3, BSE analysis images, which can observe composition-dependent contrast, make it easier to distinguish each phase. In Fig. 3, showing microstructural changes in creep specimens according to high-temperature exposure time and DC, ID locations, the γ phase with higher mass per unit volume appears bright while the γ′ phase appears dark. While the initial heat-exposed specimens show γ′ phase precipitates within the γ matrix phase, as exposure time increases, the topological inversion phenomenon becomes prominent where the precipitate γ′ phase appears in an interconnected form, and the boundaries between the two phases become blurred due to diffusion and dislocation network introduction. Although it’s clear that the material’s microstructural characteristics change with high-temperature exposure time, expressing this in quantitative descriptors (features) is challenging. Unlike other alloy systems, single crystal cast CMSX-4 alloy lacks grain boundaries, making it impossible to obtain widely known microstructural descriptors such as grain size distribution, texture, and orientation relationships. Therefore, this study connected microstructure and properties by directly using raw images in CNN instead of deriving microstructural descriptors.

Fig. 3

Microstructure evolution of aged CMSX-4 alloys during creep test. The white arrow indicates creep loading direction.

3.2 Convolution neural network model

The CNN architecture used in this study is shown in Fig. 4 and Table 4. Since the DC and ID regions of the specimen show different microstructures depending on location, as shown in Fig. 3, we constructed a network where microstructural images from each region were used as input layers and passed through 16 hidden layers with identical structures. The outputs from these layers were then combined through two flatten layers to form an output layer that predicts creep life or rupture strain.

Fig. 4

Schematic of convolution neural network of this study. Two inputs from different regions (dendritic core and inter-dendritic) are passed through separate hidden layers and merged after flattening.

Table 4 CNN architecture.

The hidden layers were constructed by alternating between two types of layers: one with a convolution layer that expands channels from the input image followed by a filter and uses Leaky ReLU (leaky rectified linear unit) as an activation function, and another that adds Max Pooling operations on the expanded channels and applies 0.25 Drop Out for preventing overfitting to the same structure. The Leaky ReLU activation function is a modified version of ReLU (rectified linear unit), designed to solve the gradient vanishing problem in negative regions. Both ReLU and Leaky ReLU are non-linear activation functions that provide non-linearity to the neural network, enabling it to learn complex patterns. While similar to ReLU, Leaky ReLU applies a small gradient to negative inputs, allowing some information to be transmitted in the negative region. In this study, we used a slope of 0.1 for negative inputs.

The 256 × 256 input images were reduced by passing through hidden layers until flattening, then transformed to form a 1 × 4092 fully connected layer from two input images. The fully connected layer was reduced to one dimension through two linear layers of the same dimension without activation functions, with the predicted values being either creep life or creep rupture strain. The loss function used for training was the mean squared error between predicted and experimental values, and Adam [26] was used as the optimization function. This neural network was implemented using Python [27] and PyTorch [28], and training was performed using a Nvidia GeForce RTX 4090 GPU with CUDA Toolkit 12.6.0.

3.3 2-points spatial correlation analysis

Two-point spatial correlation analysis is a method that analyzes the correlation between two points in space according to distance and direction, representing the probabilistic relationship between the two points. As shown in Fig. 3, by performing binary segmentation of two phases with contrast differences on a grid and calculating the phases forming the two points using eq. (1), the spatial probability can be interpreted.

  

\begin{equation} f(h,h'|r) = \frac{1}{|\Omega(r)|}\int_{\Omega(r)}m(h,x)m(h',x + r)dx \end{equation} (1)

In eq. (1), represents the correlation function between two points with distance r and phases h, h′, and represents the area (2D) or volume (3D). In 2D microstructure images like Fig. 3, h can be either γ or γ′ phase, and when h and h′ are the same, it becomes an autocorrelation. In binary segmented microstructure images, the bright phase m(γ) has a value of 1, and the dark phase m(γ′) has a value of 0. PyMKS [22], released by the Kalidindi group, is a representative Python package that can calculate two-point spatial correlations of microstructures. However, in this study, we performed direct calculations using Python Numpy [29].

Figure 5 shows the γ-γ and γ-γ′ correlations calculated within a 256 × 256 size range after binary segmentation of 8-bit grayscale BSE images acquired at 960 × 960 pixels from DC and ID regions of each specimen, as shown in Fig. 3. The original BSE images obtained at 1280 × 960 size had a resolution of 54 pixels/µm. The 256 × 256 size range was determined to include more than 4–5 γ-γ′ layers to obtain correlation data from a sufficient area in the images shown in Fig. 3. As the topological inversion phenomenon progressed, the connections between the γ matrix phase disappeared while the γ′ precipitate phase began to connect, while maintaining the longitudinal and transverse orientation relationships from single crystal solidification. The γ-γ′ correlation spectra obtained by performing the same analysis on all BSE images of the specimens were used to train the aforementioned CNN model.

Fig. 5

(a) γ-γ and (b) γ-γ′ 2-point correlations from BSE images in Fig. 3. (online color)

3.4 Creep properties predictions using CNN

The images used for CNN training are of three types: (1) 960 × 960 BSE images, (2) 256 × 256 γ-γ correlation spectra obtained through two-point spatial correlation analysis of 960 × 960 BSE images, and (3) γ-γ′ correlation spectra. 21 BSE images were acquired from the DC and ID regions of each of the 5 creep test specimens (as heat-treated and after 1000, 2000, 5000, 10000 hours of high-temperature exposure). When training the CNN with 960 × 960 BSE images, a layer that reduces the image size to 256 × 256 was added before the Conv1 layer shown in Table 4. Out of 210 images in each dataset, 150 were used for training, 30 for validation, and the remaining 30 for testing. Since it is very difficult to acquire a large number of microstructural images of materials for CNN training, it is common to increase the number of training data using data augmentation techniques [30, 31]. However, in this study, data augmentation techniques were not used because when specimens have directional characteristics, there is a risk of losing corresponding information if images are rotated or flipped.

Figure 6(a) shows the learning curve for the creep life prediction model, and (b) shows the learning curve for the rupture strain prediction model. Regardless of the type of training data, all CNN models converged within 1000 epochs. While mean-squared error (MSE) was used as the loss function for model training, Fig. 6 is displayed in root mean-squared error (RMSE) for comparison with experimental values. Figure 7 shows the creep life and rupture strain results of test data predicted by CNN.

Fig. 6

Learning curves of CNN models for (a) rupture life and (b) strain predictions. (online color)

Fig. 7

CNN results of (a) rupture life and (b) strain predictions using different input images (black: BSE image, red: γ-γ correlation, and blue: γ-γ′ correlation). (online color)

For the creep life prediction model, when using BSE images for training, the R² value and mean absolute error (MAE) were 0.952 and 12.93 respectively; when using γ-γ correlation, they were 0.766 and 20.33; and when using γ-γ′ correlation, they were 0.980 and 8.35, showing the highest performance with the model using γ-γ′ correlation. Similarly, for creep rupture strain prediction, when using BSE images for training, the R² value and MAE were 0.648 and 4.37 respectively; when using γ-γ correlation, they were 0.770 and 3.25; and when using γ-γ′ correlation, they were 0.786 and 2.72, showing the highest performance when using γ-γ′ correlation. Therefore, it can be concluded that the spatial relationship of γ and γ′ phases obtained through two-point spatial correlation analysis had an effective impact on improving the model for predicting the creep properties of CMSX-4 alloy.

Obtaining microstructural images of alloys requires significant time and cost, while simultaneously presenting challenges in deriving appropriate microstructural descriptors that can be linked to material properties. In particular, for the CMSX-4 single crystal material studied in this research, conventional microstructural descriptors widely used in other materials, such as grain size and orientation relationships, do not exist. Alternatively, in cases like composites where phases with different characteristics coexist, additional modeling methods are needed to obtain descriptors that correspond to the entire microstructural region [32]. Although developments in CNN models and computational resources have made it possible to directly train raw microstructural images, the task of acquiring microstructures from areas large enough to sufficiently represent specimen characteristics remains essential. In microstructures composed of γ and γ′ phases, two-point spatial correlation analysis provides a microstructural descriptor calculated as a two-dimensional spectrum, which conveys sufficient information for predicting material properties when trained with artificial neural networks. This has been demonstrated through this research, and it is expected that this approach can be utilized in various artificial neural network studies involving microstructural images.

4. Conclusion

In this study, a model was constructed to predict creep life and rupture strain by training microstructural images of materials using deep artificial neural networks to predict changes in creep properties of CMSX-4 alloy according to high-temperature exposure time. To predict the creep life and rupture strain of specimens exposed to high temperatures from 0 to 10000 hours, CNN models were trained using BSE images of specimens and images obtained through two-point spatial correlation analysis, and their results were compared. In CMSX-4 alloy, topological inversion occurs where the arrangement of γ and γ′ phases changes with high-temperature exposure time, resulting in significant microstructural changes. Among the CNN models constructed to predict creep properties according to microstructural evolution, which were trained using 8-bit grayscale BSE raw images, γ-γ correlation, and γ-γ′ correlation as training images, the model trained with γ-γ′ correlation showed the best performance in predicting both creep life and rupture strain.

Acknowledgments

This study was supported by the National Research Foundation of Korea (NRF-2021M3H4A6A01045764 and NRF-2021M3A7C2089767).

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