Deep learning for the rare-event rational design of 3D printed multi-material mechanical metamaterials

The rational design of architected materials with anisotropic properties is crucial for achieving optimal, multifunctional performance. Nature provides examples of such optimization, like the combination of high stiffness and light weight in wood and bone, or ultrahigh stiffness and toughness in nacre. In man-made metamaterials, specific combinations of mechanical properties are sought to enable novel functionalities, such as combining auxetic behavior with high stiffness for structural applications. The main challenge is identifying the micro-architectures that yield the desired properties. Multi-material 3D printing now allows for complex geometries and arbitrary material distributions, greatly expanding design possibilities but also the search space. Optimizing these parameters requires a massive number of computational models, particularly for rare events like double-auxeticity combined with high stiffness. This research addresses this challenge by utilizing deep learning to drastically reduce computational time and enable the exploration of a vast design space.

The literature highlights the advantages of using machine learning in various scientific fields, including composite and metamaterial design, material property prediction, and manufacturing process optimization. However, the application of deep learning to the design of multi-material mechanical metamaterials to achieve very rare target properties has not been extensively explored. Previous work has shown the rarity of double-auxeticity in 2D lattices and the challenges in combining this with high stiffness. The current study builds upon these limitations, proposing a deep-learning approach to efficiently navigate this high-dimensional design space.

This study used planar lattices based on re-entrant, cubic, and honeycomb unit cells with random distributions of hard and soft phases. The ratio of hard phase volume (p1) was varied. Finite element (FE) models generated the training dataset for a deep learning model (single unit cell model). Three designs (one from each unit cell type) were 3D printed and mechanically tested using digital image correlation (DIC) to validate the computational models. A second deep learning model (four-tile model) was trained to predict properties from combinations of four tiles. Seven additional tiled designs were 3D printed and tested. The FE simulations used three-node quadratic beam elements (Timoshenko beam elements), with different Young's moduli assigned to the hard and soft phases. Two deep neural networks were implemented using TensorFlow.keras, one for single unit cells and one for four-tile combinations. Hyperparameters were optimized using a validation dataset. Training errors were calculated using MAE and MSE. The experimental setup involved a multi-material 3D printer (Object500 Connex3) using VeroCyan™ (hard) and Agilus30™ (soft) polymers. Mechanical testing used a Lloyd instrument, and DIC was used to capture full-field strain patterns. The study also investigated stress uniformity as a design criterion.

The deep learning models accurately predicted mechanical properties, with significantly reduced computation time compared to FE simulations (≈2.4 × 10⁻⁶ s per deep learning prediction). The single unit cell model predicted a wide range of elastic moduli and Poisson's ratios. The four-tile model expanded the achievable range of properties, especially enabling a near-square region of attainable properties along direction 2. The relationship between Poisson's ratio and volume fraction of hard phase (p1) showed non-monotonic behavior, explained by the non-affinity imposed by random material distribution. Combining different unit cell types in four-tile and nine-tile structures significantly expanded the achievable properties. Experimentally validated results showed close agreement with computational predictions. The study further examined stress uniformity, identifying designs with minimal stress concentration and improved structural integrity. Combining different unit cell types showed potential to enhance functionality, such as creating shape-morphing boundaries and specific Poisson's ratio values. The research indicates the significant influence of combining different unit cells in a four-tile structure in expanding the range of attainable elastic properties. Notably, the study identified designs exhibiting double-auxeticity with high stiffness, a previously rare combination. The combination of re-entrant and orthogonal unit cells boosted the elastic modulus in the constituent re-entrant unit cell while maintaining the extreme negative Poisson's ratio. Similarly, combining honeycomb and orthogonal unit cells boosted the elastic modulus of the constituent honeycomb unit cell without significantly changing the extreme positive Poisson's ratios. Rotating a design by 90 degrees increased the elastic modulus in the weaker direction, creating more isotropic structures.

The results demonstrate the effectiveness of deep learning in accelerating the design process of multi-material metamaterials. The ability to efficiently explore a vast design space enables the identification of rare designs with desired combinations of mechanical properties, such as double-auxeticity and high stiffness. The experimental validation confirms the accuracy of the computational models and the feasibility of the proposed design approach. The inclusion of stress uniformity as a design criterion contributes to the development of more robust and reliable metamaterials. The findings contribute significantly to the field of metamaterial design and open new possibilities for the development of advanced materials with tailored functionalities.

This research successfully employed deep learning to accelerate the design of multi-material mechanical metamaterials, enabling the discovery of rare designs with desirable properties. The high accuracy of the deep learning models, validated experimentally, allows for efficient exploration of a vast design space. The approach enabled the identification of double-auxetic designs with high stiffness and the consideration of stress uniformity as a design parameter, leading to more robust structures. Future research could explore a wider range of material ratios and unit cell designs, further expanding the potential of this design methodology.

The study considered a fixed ratio of elastic moduli between the hard and soft phases. Varying this ratio could further broaden the range of attainable properties. The focus on planar lattices limits the generalizability to 3D structures. While stress uniformity was considered, a more comprehensive investigation of failure mechanisms could enhance design robustness further. The range of materials used in 3D printing could also be expanded to explore more diverse material combinations and functionalities.