Seismic waves damping with arrays of inertial resonators

We investigate stop band properties of a theoretical cubic array of iron spheres connected to a bulk of concrete via iron or rubber ligaments. Each sphere moves freely within a surrounding air cavity, but ligaments couple it to the bulk and facilitate bending and rotational motions. Associated low frequency local resonances are predicted by an asymptotic formula. We find complete stop bands in the range [16,21][16,21] Hz (resp. [6,11][6,11] Hz) for 7.4 m (resp. 0.74 m) diameter iron spheres with a 10 m (resp. 1 m) center-to-center spacing, when they are connected to concrete via steel (resp. rubber) ligaments. The scattering problem shows that only bending modes are responsible for damping: rotational modes are totally overwritten by bending modes. Regarding seismic applications, we consider soil as a host medium, in which case the low frequency stop band can be enlarged through ligaments of different sizes that allow for well separated bending and rotational modes. We finally achieve some damping of elastodynamic waves from 8 to 49 Hz for iron spheres 0.74 m in diameter, connected to soil with six rubber ligaments of optimized shapes. These results represent a preliminary step in the design of seismic shields placed around, or underneath, foundations of large civil infrastructures.