Multiphase alloys and metal-matrix composites constitute an important class of structural materials (Wang and Hwang, 1998; Tasan et al., 2015; Latypov et al., 2016). The presence of two or more phases allows unique combinations of properties inaccessible to single-phase materials. Accelerated design of multiphase materials and their process optimization could benefit from efficient computational tools that can accurately capture the relationships between mulitphase microstructure and overall engineering properties.

In this context, a number of modeling approaches and strategies have been developed to date. Historically earliest, the Voigt and Reuss models (Voight, 1928; Reuß, 1929) estimated the overall or effective properties of composites from the volume fractions of their constituents. These models are based on the assumptions of either uniform stress or uniform strain throughout the composite and serve as the lower and upper bounds of the effective property of interest (e.g., stiffness or strength). While straightforward, these bounds are often widely separated, especially in high-contrast composites with a large difference in the properties between the constituents (Latypov et al., 2019a). Tighter bounds for effective elastic properties were obtained by Hashin and Shtrikman for composites with moderate elastic contrasts (Hashin and Shtrikman, 1963). Self-consistent models also offer better estimates of effective properties compared to bounds. Most of these approaches, however, are based on volume fractions of the constituents, which limits their use for microstructure-sensitive design of composite materials and structural components (Fullwood et al., 2010).

With the emergence of high-performance computing, computational homogenization has become a viable and powerful approach to calculating effective properties of heterogeneous materials (Segurado et al., 2018). The finite element (FE) methods (Gilormini and Germain, 1987; Ghosh et al., 1995; Segurado and Llorca, 2002; Latypov et al., 2019a) and fast Fourier solvers (Michel et al., 1999; Lebensohn, 2001; Eisenlohr et al., 2013; Lucarini et al., 2021) both allow calculating effective elastic and inelastic properties with the direct account for three-dimensional (3D) microstructure. The microstructure effects are explicitly incorporated by performing calculations on 3D representative volume elements (RVEs) or microstructure volume elements (MVEs) of the multiphase microstructure of the material. The account for microstructure is associated with a significant increase in the computational cost compared to analytical calculations of bounds and faster numerical self-consistent methods discussed above. The computational cost of RVE-based methods is still prohibitively high for routine use in multiscale modeling of commercial metal forming processes.

Surrogate or reduced-order models have been proposed as a strategy to address the trade-off between the computational cost and incorporation of 3D microstructure effects. Materials Knowledge Systems (MKS) is an example computational approach of surrogate model development (Gupta et al., 2015; Latypov and Kalidindi, 2017). In MKS, 3D microstructures are quantified using the n-point spatial statistics with subsequent establishment of quantitative microstructure–property relationships in the form of polynomial functions fitted to data from numerical (e.g., FE) simulations. Other forms of microstructure—property relationships besides polynomial functions are also seen in literature, including statistical learning models [e.g., Gaussian process regression (Marshall and Kalidindi, 2021)], or neural networks, including convolutional neural networks, CNN (Cecen et al., 2018; Yang et al., 2018; Ibragimova et al., 2022; Mann and Kalidindi, 2022) and graph neural networks (Dai et al., 2021; Hestroffer et al., 2023). Owing to the computational efficiency and account for 3D microstructure, surrogate models offer a promising pathway towards practical implementation and industrial adoption of microstructure-sensitive multiscale models of full-scale metal forming processes.

Both the computational homogenization methods and their data-driven surrogates require three-dimensional (3D) microstructure data in the form of an RVE as input for property calculations. The RVE needs to have sufficiently high resolution and sufficiently large size to capture both the 3D morphology and spatial distribution of the microstructure constituents (continuous phases, particles, or grains). Advances in experimental characterization have made it possible to obtain 3D microstructure datasets for a wide variety of materials. Experimental 3D datasets are typically obtained from serial sectioning (Groeber et al., 2006; Echlin et al., 2012) or directly using non-desctructive 3D characterization techniques (Pokharel et al., 2015; Shahani et al., 2020). Most of the experimental techniques that yield high-quality 3D microstrcuture data require access to high-cost and unique experimental facilities. Consequently high-quality 3D experimental data are currently scarce in the materials research community, especially in industry. An alternative to 3D characterization is digital generation of 3D microstructure RVEs or MVEs based on 2D microstructure data (Bostanabad et al., 2018). For example, Dream.3D is an open-source software commonly used for 3D microstructure generation based on statistical distributions of sizes, shapes, and volume fractions of the microstructure constituents (Groeber and Jackson, 2014). Dream.3D has been widely used for building RVEs as input for microstructure-based simulations (Latypov and Kalidindi, 2017; Diehl et al., 2017). Recently, new approaches have been emerging for 3D RVE generation from 2D microstructure data, including a 3D generation algorithm inspired by solid texture synthesis in computer graphics (Turner and Kalidindi, 2016) or transfer learning technique (Bostanabad, 2020).

In this work, we propose a machine learning framework that relates the microstructure from 2D sections directly to effective 3D properties of heterogeneous materials and specifically two-phase composites. The framework is based on a hypothesis that 2D microstructure data encapsulates sufficient statistical information to capture the effective properties of heterogeneous media to a satisfactory extent. We explore the application of this framework to modeling effective mechanical properties of two-phase materials using two approaches: 1) statistical learning with spatial correlations as microstructure descriptors and 2) deep learning with CNNs as microstructure feature extractors. Section 2 describes these two approaches followed by their application on stiffness of two-phase microstructures presented and discussed in Sections 3–5.

The key distinguishing feature of the computational framework developed in this study is the extraction of quantitative microstructure features from 2D orthogonal microstructure sections, their aggregation and use for machine learning models for effective properties. To this end, we develop and critically compare two approaches. In the first approach, which we refer to as the Multisection MKS approach, we use n-point correlation functions and principal component analysis (PCA) to engineer microstructure features from 2D orthogonal sections. In the second, Multisection CNN approach, we train CNNs as feature extractors for microstructure images. In both cases, features extracted from individual 2D sections are combined into a single feature vector for establishing quantitative relationships between microstructures and effective properties of interest.

The key results of the case study with the two new models—Multisection MKS and Multisection CNN—are shown in the form of parity plots in Figures 4B,C. The performance of the two models is compared to the benchmark 3D MKS model (Figure 4A). The parity plot depicts predictions of the developed machine learning models against the ground-truth FE simulation results. Each point represents a prediction and ground truth result for one MVE; a perfect agreement would be seen as points lying on the parity line. All the models produce a reasonable agreement between the trained machine learning models and the FE simulations. Indeed, the predictions of all three models show a reasonable homoscedastic distribution along the parity line. The Multisection MKS model predictions are characterized by a wider scatter compared to the 3D MKS model, which is the “price” of relying on 2D microstructure information rather than full 3D data. The Multisection CNN model also shows a wider scatter compared to the benchmark model but to a smaller extent than the Multisection MKS model. The quantitative comparison of the MAE normalized by ground truth values supports these observations: the Multisection MKS model has the highest MAE of 26.6% vs. 13.1% of the benchmark 3D MKS model, while the MAE of the Multisection CNN is 19.2% (all for the testing set).

The results of the case study presented above show the feasibility of machine learning models that relate effective properties of two-phase microstructures directly to 2D microstructure sections. The value of the developed Multisection models arises from the prevalence of 2D microstructure data in the materials community, especially in industry, compared to scarce 3D data. The transition from full 3D microstructure data to 2D sections used for the MKS model development comes at the price of about 13% loss in accuracy. The trade-off between the completeness of the microstructure information and the accuracy can be addressed by more advanced models such as CNNs: we report only a 6% increase in MAE with the Multisection CNN compared to the MKS model based on full 3D microstructure information. This sacrifice in accuracy can be reasonable in many use case scenarios given the significant savings in time, labor, and resources needed to collect 2D section data compared to 3D characterization. An additional advantage of the best performing Multisection CNN is manifested in the computational cost of training: it takes only about 12 min to train or Multisection CNN with CPUs (hardware specified in Section 3.1.2), while previous studies reported hours of training for 3D CNNs on the same dataset of high-contrast composites (Cecen et al., 2018). The Multisection CNN model developed here is thus a light-weight data-driven model which strikes a balance between minimal requirements in terms of microstructure input (2D sections), decent accuracy, and modest computational resources and time needed for training.

The Multisection models presented here show a viable strategy of modeling effective properties of heterogeneous materials without the need of experimental 3D microstructure data or an extra computational step of reconstructing 3D micrsotructures from 2D data. The models trained on 2D microstructure sections from 3D synthetic microstructures used in the simulations can be leveraged for predicting properties of materials based on experimental microstructure images. While we demonstrated the predictive capabilities of the Multisection framework on effective elastic properties, it can be readily extended to effective inelastic properties of multiphase materials (Latypov and Kalidindi, 2017) as well as elastic and inelastic properties of polycrystalline materials (Paulson et al., 2017). Indeed, CNNs can be configured to take crystal orientations as input [e.g., (Pandey and Pokharel, 2021)], while calculation of two-point correlations for polycrystalline MVEs within the MKS framework is also available using generalized spherical harmonics (Paulson et al., 2017) as the microstructure description basis. Finally, in this study, we trained a relatively simple architecture from scratch to test the concept, however, use and fine-tuning of existing state-of-the-art models (such as ResNet) as the base CNN architecture for each 2D section can futher improve the results and accuracy of the Multisection CNN approach.

We presented a new Multisection machine learning framework for predicting effective properties of heterogeneous materials. The framework is based on the hypothesis that the microstructure data from 2D sections suffice for establishing a quantitative relationship between the microstructure and the effective property of interest. We tested the hypothesis in a case study of predicting effective stiffness in high-contrast two-phase composite microstructures. To this end, we tested two computational approaches: Mutisection MKS and Multisection CNN. Both approaches rely on aggregating microstructure features extracted from three 2D sections. In the case of Multisection MKS, we obtain microstructure features by principal component analysis of two-point correlation functions calculated for 2D sections. In the case of Multisection CNN, we utilize CNN filters as feature extractors with subsequent pooling of features from individual 2D sections. We show that both the Multisection MKS and Multisection CNN models show reasonable accuracy with 13% and 6% increase in the mean average error compared to the previously published MKS model relying on full 3D microstructure data. Our results confirm the hypothesis and demonstrate the Multisection CNN approach as an optimal model for predicting effective properties of heterogeneous media without the need to collect or reconstruct 3D microstructure volumes. The Multisection CNN is therefore a powerful addition to the computational materials science toolkit for modeling microstructure—property relationships.