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In an attempt to model the behavior of microstructurally small cracks, both empirical and numerical approaches have been employed based on certain underlying principles and influencing factors. Fatemi and Yang8 and Hussain5 review early phenomenological theories on small-crack behavior, in which plasticity effects, metallurgical effects, and crack closure are suggested as possible explanations for small-crack behavior. Factors such as grain shape and crystal orientation, near-neighbor distances, grain fracture toughness, and intrinsic flaw size are also considered as factors contributing to crack nucleation and early-stage propagation.9,10 In some cases, finite-element analysis using crystal plasticity models, as originally proposed by Asaro,11 are used in literature to investigate the effect of the spatial variability of microstructural features and micromechanical fields on small-crack behavior (e.g., see Ref. 12). A more comprehensive review of empirical and numerical approaches investigating the behavior of small cracks can be found in other literature.5,6,7,8,13

While the aforementioned approaches often rely on stress intensity factors12 and/or fatigue indicator parameters14,15 as representative mesoscopic surrogates for the driving force behind crack initiation and propagation, there is a need for a more comprehensive, general framework that accounts for the complex spatial, nonlinear relationships between the relevant features, i.e., microstructural features and micromechanical fields, and a given representation of the crack. Data-driven methods can potentially be a viable alternative approach to address this challenge, as they can leverage large, high-dimensional datasets, obtained through experiments or simulations, to model these complex relationships.16,17,18 Machine learning (ML) models have already exhibited a wide range of applicability within the materials science community, including for materials discovery,19 optimal design of experiments,20 and image-based materials characterization.21,22 In the context of predicting crack behavior, data-driven methods such as principal component analysis (PCA) have been employed to determine reduced-order representations that correlate with fatigue indicator parameters (FIPs).23,24 In similar work,25 a random forest learning algorithm was employed to predict stress hot spots that were computed from full-field crystal plasticity simulations, where the algorithm was trained using features that encode the local crystallography, geometry, and connectivity of the microstructure. Probabilistic models such as Bayesian networks,7,26 which are nonparametric and can account for uncertainties in predictions, have also been used to compute fatigue-related parameters such as residual life and equivalent stress intensity factors, respectively.

This paper expands previous work on data-driven methods in fatigue modeling and proposes a framework using a convolutional neural network (CNN) model to predict 3D crack paths based on microstructural and micromechanical features. While previous, relevant work using data-driven approaches has primarily focused on identifying micromechanical and microstructural variables that contribute to the direction and rate of crack propagation,26 or on mapping global variables (such as chemical composition, grain size, heat treatment, and cyclic stress intensity factor) to the one-dimensional crack growth rate,27 the work presented herein concerns the use of CNN to quantitatively predict the local crack path, in 3D, as a function of local microstructural and micromechanical features. CNNs are particularly well suited for problems that require finding spatial, nonlinear relationships between input and a given response variable of interest and have been successfully used in related applications such as classification of microstructures based on scanning electron microscopy (SEM) images20 and determination of material properties based on microstructure.28,29 Prior to training the CNN model, the input features (i.e., microstructural features and micromechanical fields) are selected based on previous correlation analysis by Pierson et al.30 (described below). PCA analysis is performed to convert the relevant input features to unique low-dimensional descriptors, whose values are specific to a given location within a microstructure. The 3D map of these descriptor values is then introduced into the CNN model. As a postprocessing step, to retain spatial continuity, a smoothing operation is performed on the predictions of crack surface elevations obtained from the CNN model.

In previous work,30 the authors conducted a systematic correlation analysis between computed micromechanical fields in a 3D, uncracked polycrystal and the observed path of an eventual fatigue crack. Specifically, an experimentally measured volume of an Al-Mg-Si alloy31 was modeled using a high-fidelity, concurrent-multiscale, finite-element mesh with a crystal-plasticity constitutive model.32 Cyclic loading was simulated at a load ratio of R = 0.5 (consistent with experiment), and computed field variables (or derivatives thereof) based on stress, strain, and slip were parameterized to a regular 3D grid. A complete list of the 22 variables computed at each time step during the finite-element simulation is presented in Table I. Additionally, the cyclic change in each of the variables was computed between the peak and minimum load for each of five simulated loading cycles. Figure 1 shows six of the parameterized variables (five cyclic damage metrics and the cyclic micromechanical Taylor factor33). Once the cyclic change in the computed field variables was shown to converge, the spatial gradients of the variables were calculated using a finite-difference approach. In total, 88 variables were considered in the correlation study, of which 44 were based on spatial gradients of the micromechanical fields. The parameterized variables were then systematically correlated with distance to the known crack surface. Correlation coefficients for all 88 variables are shown in Fig. 2. In general, the spatial gradients of the micromechanical field variables (Fig. 2c and d) exhibited a stronger correlation with the crack path than did the field variables themselves (Fig. 2a and b). To assess whether the correlation coefficient values shown in Fig. 2c and d were meaningful, correlation analyses were also performed between the 88 variables and alternative paths throughout the microstructure. The correlation coefficients for the alternative paths were consistently weaker than those for the actual crack surface, suggesting that micromechanical fields of the cyclically loaded, uncracked microstructure might provide some degree of predictiveness for the microstructurally small fatigue crack path. Results from the previous correlation analyses are leveraged in the current work involving physics-informed ML to predict the crack surface evolution.

Grid data showing cyclic values of slip-based damage metrics computed for uncracked microstructure in multiscale finite-element simulation. A description of each metric is provided in Table I. Reprinted with permission from Ref. 30

Based on the previously described correlation analysis, a new, low-dimensional representation of the data is computed from the original set of features, one that is amenable to spatial relation-based learning algorithms, without losing high-value information contained within the raw features. To achieve this, the existing data are transformed into a new domain that has a low dimensionality for each point in the 3D microstructure. As an analogy, consider how an image is represented in a low-dimensional color space (such as RGB) and contains a vector with three entries at each of the pixels on a two-dimensional (2D) plane. This representation allows for feasible convolutions and other operations during training. In this work, the dimensionality of each point within the microstructure is first reduced from 88 raw features to a descriptor vector containing three elements, while still retaining the high-value information for predicting the crack path.

In addition to the features based on micromechanical fields, an additional feature is considered based purely on the geometrical configuration of the microstructure. Experimental evidence has long suggested that microstructurally small fatigue cracks behave very differently near grain boundaries than within a single grain, manifested as crack deceleration or deflection.35,36,37 To account for such a spatially dependent relationship, an additional feature called \(d_{\rm GB}\), which is the distance from a given point to the nearest grain boundary, is added to the descriptor set to produce a location-specific, four-element descriptor. Thus, the descriptor vector \(\mathbf {{x}_{i}}\) corresponding to each location i can be expressed as follows: \({\mathbf {{x}_{i}}} = [\alpha _{1i}, \alpha _{2i}, \alpha _{3i}, {d}_{{\rm GB},i}]\). For comparison, the ML models are also trained using a descriptor vector of only the experimentally measured Euler angles (represented within the fundamental zone, to account for crystal symmetry) corresponding to each location, to see how predictions based on a simpler, more rudimentary description of the microstructure perform in comparison with the feature vector \(\mathbf {{x}_{i}}\). 2b1af7f3a8