Hierarchical design of mechanical metamaterials: an application on pentamode-like structures

• Introduces and demonstrates a Hierarchical Metamaterial Automated Design (H-MAD) framework that generates multiscale designs. • H-MAD overcomes large-scale direct design condiguration encoding challenges to navigate a complex design space of 15 ¹⁰⁰ configurations. • Optimizing structural connectivity and property distribution are key to hierarchical microstructure selection and placement trade-offs. • A case study produces pentamode-like structures that achieve B/G > 10,000, outperforming classic and non-hierarchical pentamodes.. • Consistent pentamode-like behavior is observed across many designs, indicating resilience to fabrication errors and material uncertainties. Mechanical metamaterials demonstrate that unprecedented static and dynamic behaviors can emerge from engineered nonhomogeneous architectures. However, most designs operate at a single length scale and optimize a single performance criterion. Evidence from nature and prior studies suggests that multiscale architectures can enhance performance and broaden applications, yet their design remains challenging. To address this, a Hierarchical Metamaterial Automated Design (H-MAD) framework has been developed. This framework employs a sequential, evolutionary optimization approach to generate a population of heterogeneous structures at each scale, optimize microstructure placement and ensure cross-scale compatibility. As a case study, H-MAD is applied to 2D pentamode-like hierarchical metamaterials designed for extreme bulk-to-shear modulus ratios ( B / G ). The resulting architectures exhibit pentamode-like behavior across diverse configurations, demonstrating the framework's efficacy. With just two length scales, these hierarchical designs surpass both non-hierarchical structures with the same mesoscale configuration and classical pentamodes with finite joints. The best design achieves B/G ≈ 15 × 10 ³ under bulk modulus constraints—nearly an order of magnitude higher than the baseline pentamode ratio. Even without constraints, the optimal H-MAD design attains B/G > 10 × 10 ³ , significantly outperforming conventional pentamodes. The results demonstrate that hierarchical design, combined with stiffness tailoring across scales, can enhance mechanical performance while maintaining adequate bulk moduli. The persistence of pentamode-like performance across diverse hierarchical designs indicates resilience to fabrication imperfections and material uncertainties, ensuring robust performance in practical applications. This advancement in hierarchical metamaterial design represents a step towards expanding the limits of metamaterial mechanical performance and applicability in various engineering domains.