Hierarchical metamaterial with auxetic foams for low-frequency seismic attenuation

Seismic metamaterials (SMs) show immense potential for vibration mitigation, yet challenges remain in achieving ultra-low-frequency and broadband attenuation. This paper proposes a hierarchical SM design incorporating auxetic foams with negative Poisson’s ratio (NPR), capable of attenuating both Lamb and surface waves. For Lamb wave attenuation, the hierarchical NPR-based SM is developed and analyzed using the finite element method in conjunction with the k ( ω ) technique. Computed complex band structures reveal the presence of evanescent modes, downward-shifted band gaps, and additional band gaps induced by hierarchical resonances. For surface wave attenuation, a second SM configuration integrates a square section within the NPR-based hierarchical unit cell, enabling the emergence of negative effective mass and the formation of continuous, ultra-low-frequency band gaps below 29 Hz. Finite element simulations verify the attenuation performance in both frequency and time domains. The analysis further elucidates how hierarchical arrangement facilitates multi-scale resonances and how the negative effective mass mechanism suppresses surface wave modes within attenuation zones. Time-domain simulations under seismic excitations from the El Centro and Taft records demonstrate substantial ground motion attenuation, achieving RMS acceleration reductions of 90.93% and 84.12%, respectively, at the top of a 10-story steel frame structure. Laboratory-scale experiments further validate these numerical results, confirming a superior suppression performance enabled by the negative effective mass effect. These findings highlight the practical potential of hierarchical NPR-based seismic metamaterials in enhancing the seismic resilience of large-scale infrastructure.