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

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 approximate to 15 x 10<SUP>3</SUP> 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 x 10<SUP>3</SUP>, 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.