Engineered metabarrier as shield from seismic surface waves

Resonant metamaterials have been proposed to reflect or redirect elastic waves at different length scales, ranging from thermal vibrations to seismic excitation. However, for seismic excitation, where energy is mostly carried by surface waves, energy reflection and redirection might lead to harming surrounding regions. Here, we propose a seismic metabarrier able to convert seismic Rayleigh waves into shear bulk waves that propagate away from the soil surface. The metabarrier is realized by burying sub-wavelength resonant structures under the soil surface. Each resonant structure consists of a cylindrical mass suspended by elastomeric springs within a concrete case and can be tuned to the resonance frequency of interest. The design allows controlling seismic waves with wavelengths from 10-to-100 m with meter-sized resonant structures. We develop an analytical model based on effective medium theory able to capture the mode conversion mechanism. The model is used to guide the design of metabarriers for varying soil conditions and validated using finite-element simulations. We investigate the shielding performance of a metabarrier in a scaled experimental model and demonstrate that surface ground motion can be reduced up to 50% in frequency regions below 10 Hz, relevant for the protection of buildings and civil infrastructures.

The vertical and horizontal displacement components and of the wave field relate to the potentials Φ and H y as follow: We study the dynamic of a three degree of freedom resonator subjected to the base excitation induced by the horizontal and vertical displacements 0 = ( = 0) and 0 = ( = 0), respectively, generated by a surface Rayleigh wave (Fig S1(a) and S1(b)). The resonator has three-degree-of-freedom namely the horizontal and vertical displacements of its mass with respect to the base of the resonator, and , respectively, and θ the angle of rotation of the mass with respect to the axis. The translational mass is denoted by whereas the rotational inertia of the mass as . We assume that the vertical motion is uncoupled from both horizontal and rotational displacements, while the horizontal and rotational motions are coupled when the bearings are not symmetric, i.e. = ℎ,1 being the horizontal stiffness of the bottom and top elastic bearing, ℎ,1 , ℎ,2 respectively (see Fig. S1a) Therefore, the equations of motion of the resonator subjected to the base displacements 0 and 0 read: where is the stiffness along the vertical direction; ℎ = ℎ,1 + ℎ,2 = ℎ,2 (1 + ) is the total stiffness along the horizontal direction; = ℎ,2 ℎ 2 (1+ ) is the rotational stiffness and ℎ = ℎ = ℎ,2 (1− )ℎ 2 the coupled rotational-horizontal stiffness.
For the vertical motion, we assume a wave solution of the form = 0 ( − ) and substitute it into Eq. (s.3a) to obtain 2 : Similarly, the horizontal-rotational Eqs. (s.3b) can be uncoupled by means of modal analysis, as: where is the natural frequency, the eigenmode, the modal displacement and the modal participation factor. By assuming a wave solution = ,0 ( − ) for the resonator mode, and substituting it into Eq.(s.5) we obtain: The horizontal and rotational displacements are thus obtained as: This coupling arises from the elliptical motion of surface waves.
In Fig. S1(c) we show the real roots of the dispersive relation ( ) for the soil coupled with the 3 dof resonators (the geometrical and mechanical parameters in Table S1 of the main text have been used). For this symmetric resonator, the two resonant modes are uncoupled and correspond to a pure horizontal translation and a pure rotation of the resonator with eigenfrequencies: However, in a full uncoupled system the rotational motion cannot be excited by the horizontal base displacement. On the contrary in the real system, the variation of the surface wave horizontal displacement along the resonator depth induces resonator rotation. To account for this we introduce a small numerical asymmetry = 1.01. As shown in Fig. S1(c) the rotational motion does not substantially effect the position and size of the occurring band gap. However, we observe a further flat branch at the rotational resonance frequency.
We underline that the use of the modal analysis to extract the response of the resonator can be extended to multi-dof (or even multi-mass) resonator systems. Corresponding dispersion relation.

S2 BAND GAP FREQUENCY EDGES AND NORMALIZED BANDWITH
The approximate dispersion relation for the soil-resonator system obtained by neglecting the horizontal and rotational resonances reads (2):

S2. Experimental characterization of the resonator
In Fig. S2(a) we show a typical transient response of the resonator to incoming surface waves.
The resonator shows a narrow spectrum centered around its resonance frequency fv,exp (Fig S2(b)).
We found a small variation in resonance frequency and quality factor over all resonators (approximately 10%).