Analytical Modeling for Nonlinear Seismic Metasurfaces of Saturated Porous Media

Abstract

Background: Rayleigh waves transport the greater part of the energy released by shallow earthquakes and dominate the damage potential of near-surface ground motion. Sub-wavelength resonant metasurfaces offer a route to intercepting this energy before it reaches a protected structure, yet almost all available closed-form treatments idealise the substrate as a dry, linearly elastic half-space.
Problem: Urban sites requiring protection are frequently underlain by shallow water tables, so the substrate is a two-phase medium in which solid–fluid coupling, pore-pressure diffusion and viscous Darcy flow act simultaneously. Under strong shaking the skeleton additionally leaves the small-strain regime, the shear modulus degrades and excess pore pressure accumulates, so both the substrate impedance and the resonator tuning drift during the event.
Method: This work develops a closed-form dispersion theory for a locally resonant metasurface bonded to a Biot fluid-saturated poroelastic half-space. Skeleton nonlinearity is introduced through a strain-compatible Hardin–Drnevich modulus-degradation law coupled to an effective-stress reduction driven by the pore-pressure ratio, and resonator nonlinearity through a Duffing restoring force treated by harmonic balance. The resonator array enters as a frequency-dependent effective surface mass, and a surface flow-resistance parameter spans the fully drained to fully sealed hydraulic limits. The resulting 3×3 boundary-condition determinant is solved for the complex wavenumber k(ω)=k_r+iα.
Results: For a benchmark saturated sand (V_S=150 m/s, Biot characteristic frequency 4244 Hz) a 5 Hz metabarrier produces 92.9 % insertion loss over a 30 m array. Strain-compatible softening to G/G_max=0.267 shifts the peak attenuation from 5.0 Hz to 4.5 Hz, a detuning that a five-stage graded array recovers, doubling the effective bandwidth from 1.25 Hz to 2.50 Hz. A modal energy balance, closing to within 0.3 %, shows that pore-fluid viscous dissipation contributes only 1.1 % of the attenuation at the design permeability but 49 % when the Biot characteristic frequency falls into the seismic band. Peak dynamic pore pressure beneath the barrier is 3.7 times the free-field value at equal surface motion.