Designing bioinspired brick-and-mortar composites using machine learning and statistical learning

The study addresses how to efficiently explore the vast design space of bioinspired brick-and-mortar (nacre-like) composites to achieve improved mechanical performance, particularly tensile strength and favorable failure modes. Traditional finite element analysis (FEA) and experiments are computationally and experimentally expensive for large design spaces. The authors propose combining machine learning (ML) and statistical analysis (SA) with FEA to identify key geometrical features governing failure modes and strength, define good versus bad design regions, and rapidly screen designs. The motivation stems from nature’s optimization (e.g., nacre achieving ~40x toughness increase) and the practical need to design metal-ceramic brick-and-mortar composites with high strength/toughness while minimizing computational cost.

The paper reviews applications of ML/SA in materials and composites design: statistical learning for biocomposite tensile strength optimization (Alakent and Soyer-Uzun); integrated FEM/MD/ML studies on constituent effects assuming brittle behavior (Chen and Gu); perceptron and CNN models predicting toughness/strength in 2D composites and identifying optimal geometries (Gu et al.); ANN for predicting unconfined compressive strength of cemented paste backfill (Qi et al.); ML for compression strength of heat-treated woods (Tiryaki and Aydin); and deep learning for structure-property linkages and effective stiffness prediction in high-contrast composites and 3D microstructures, where CNNs can outperform classic physics-based methods but may struggle in mid-range stiffness values (Yang et al.). These works demonstrate ML’s capability to map microstructure to properties and guide design, motivating its application to brick-and-mortar architectures.

- System and assumptions: A 2D brick-and-mortar composite was modeled with brittle ceramic bricks and ductile metal mortar, all assumed linearly elastic. The mortar-to-brick elastic modulus and strength ratios were set to 0.5 (typical metals vs ceramics). Poisson’s ratio was 0.3 for both. Failure initiation criteria: von Mises stress for yielding in metal (mortar) and maximum normal stress for elastic limit in ceramics (bricks). - Representative volume element (RVE): The smallest repeating unit contains two staggered bricks separated by horizontal mortar (HI) and connected by vertical mortar (VI). Periodic boundary conditions imposed: symmetric constraints on left and bottom edges; top and right edges constrained to have equal perpendicular displacements; a point force applied to the right edge to generate uniform tensile stress. The RVE does not handle shear loading and neglects edge effects. - Geometrical features and variables: Independent, dimensionless features: AR (brick aspect ratio = w/h), TR (log thickness ratio = log(t1/t2) between vertical and horizontal mortar thicknesses), vm (mortar volume fraction), and s (stagger shift normalized by RVE length). Unknowns t1, t2, w, h, and WR are obtained by solving relations among features and geometry (including vm = 2(t1 + t2)h; AR = w/h; TR = log(t1/t2); WR = 2(h + t2); LR normalized to 1). Feature bounds: AR ∈ [2, 50], TR ∈ [−2, 2], vm ∈ [0.01, 0.99], s ∈ [0.01, 0.5]. - Data generation: ~20,000 geometries were generated via MATLAB and analyzed using ABAQUS FEA. Under uniaxial tension, maximum stresses in VI, HI, and bricks were compared to respective strengths to determine the first failure mode. Designs were labeled: good (1) if VI fails first; bad (0) if HI or brick fails first. For good designs, strength was defined as the reciprocal of the maximum VI stress (under unit applied load), representing higher composite strength for lower VI stress. - Dataset partitioning: The dataset D was split into training (70%), validation (10%), and test (20%). - Statistical analysis and feature selection: Distributions of AR, TR, s, and vm across classes were analyzed. AR and TR showed separable distributions between good and bad classes; s and vm did not. Classification focused on AR and TR to reduce complexity while maintaining performance. - ML models: • Classifier: Decision tree (criterion: entropy; min samples per leaf: 153; inputs: AR and TR). Depth was tuned using validation AUC for depths 1–11; optimal depth = 6 (AUC ≈ 0.95). The classifier partitions AR–TR space into good/bad regions. • Regressor: Support Vector Regression (SVR) with RBF kernel (C = 2000, gamma ≈ 0.1525, epsilon = 0.001) trained on good designs to predict normalized strength. Hyperparameters selected via validation mean squared error (MSE). - Performance/ranking evaluation: The SVR-predicted strengths were compared to FEA-derived strengths, including rank-order comparisons on training and test sets. - Large-scale screening: After training, one million random geometries were generated. The decision tree filtered to good designs; SVR predicted their strengths. Contour maps explored pairwise feature-strength relations, acknowledging inherent dependence on all four features.

- Feature importance and class boundary: • AR (brick aspect ratio) and TR (log thickness ratio between vertical and horizontal mortar) are the dominant features distinguishing good vs bad designs; s (stagger) and vm (mortar volume fraction) show little class separation. • Box plots indicate >75% of good designs have TR < ~1.3 and AR < ~18. • Decision tree classifier (depth 6) achieves AUC ≈ 0.95 on validation and test; partitions AR–TR space with ~83% labeled as good; in the training data, ~75% of designs are good. - Failure mode statistics and locations: • In training data: ~75% VI failure (good), ~5% HI failure, ~20% brick failure (bad). • Brick failure initiates above/below the VI in ~94% of brick-failure cases due to stress concentration; ~85% of HI failures start near brick corners. - Strength prediction and ranking: • SVR with RBF kernel (C=2000, gamma≈0.1525, epsilon=0.001) yields low MSE on validation; rank-order agreement between ML and FEA is high for both training and test sets (scatter near y=x). • On Dtest, average normalized strength ≈ 0.74; top 100 designs ≈ 0.99. - Optimal geometric trends for high strength: • High TR (thicker VI relative to HI) and low vm tend to increase strength; many top designs have TR > ~1.6, vm < ~0.3, and AR between ~10–20. • In broader screening, points with relative strength > 0.9 often have AR ~30–35 or ~40–45 with vm < 0.5 and TR ≈ −1 (noting dependence on all features). - Mechanics insights: • For fixed AR, increasing TR thickens VI and generally increases strength, but may shift failure mode to HI or brick depending on AR. • For TR > 1, as AR increases from ~2, HI failure can govern; for AR > ~23, brick failure can precede HI due to elevated brick stress. - Computational efficiency: • FEA for 20,000 models took ~50 hours; ML predicted strengths for 1,000,000 geometries in ~6 seconds on the same resources.

The ML+SA+FEA framework addresses the core question of identifying robust design rules for nacre-inspired brick-and-mortar composites by learning from a large but tractable simulated dataset. By focusing on AR and TR for classification, the approach isolates key geometric drivers of failure mode and strength. The decision tree delineates regions in AR–TR space where VI fails first (desired toughening mechanism), enabling rapid filtering of designs. The SVR regressor then ranks good designs by predicted strength, closely matching FEA-based rankings, demonstrating that the learned model captures underlying mechanics sufficiently to guide design without solving FE models. The results clarify how competing failure mechanisms trade off with geometry: thicker VI (higher TR) increases VI strength, but for long bricks (high AR) the load transfer elevates stresses in HI and bricks, causing unfavorable HI or brick failures. Low vm and adequately high TR together promote strength, while AR must be moderated to maintain desired failure sequences. These findings are consistent with existing analytical insights and extend them across a much larger parametric space, offering practical design maps for materials engineers.

The study introduces a data-driven workflow combining FEA, statistical analysis, and machine learning to design brick-and-mortar composites. From ~20,000 simulated geometries, AR and TR emerged as the primary features distinguishing good (VI-failure-first) from bad designs. A depth-6 decision tree efficiently partitions AR–TR space, and an RBF-SVR accurately ranks designs by strength. Applying the trained models to one million random geometries revealed clear trends: high TR and low vm favor higher strength, and appropriate AR ranges help maintain favorable failure modes. The approach delivers orders-of-magnitude speedups in screening while preserving high fidelity to FEA trends, enabling rapid exploration of vast design spaces. Future work should address additional mechanical properties (e.g., fracture toughness, stiffness–strength trade-offs), post-yield behaviors, three-dimensional effects, and different loading conditions (e.g., shear, multiaxial). Integrating more realistic material models and manufacturing constraints could further refine design guidelines.

- Loading and boundary conditions: The RVE with periodic boundary conditions is tailored to uniaxial tension; it cannot directly handle shear or other loading types, and it neglects edge effects. - Material behavior: Linear elasticity is assumed with simple yield/fracture initiation criteria; post-yield, fracture processes, and toughening mechanisms are not explicitly modeled. - Dimensionality and idealization: The model is 2D with perfectly bonded interfaces and idealized geometry; real composites may exhibit interfacial defects, 3D effects, and processing-induced variability. - Generality of trends: Identified trends depend on the chosen feature bounds and property ratios (modulus and strength ratio set to 0.5); different material pairs or ranges may shift optimal regions.