  We investigate the elastic stop band properties of a theoretical cubic array
of iron spheres connected to a bulk of concrete via iron or rubber ligaments.
Each sphere can move freely within a surrounding air cavity, but ligaments
couple it to the bulk and further facilitate bending and rotational motions.
Associated low frequency local resonances are well predicted by an asymptotic
formula. We find complete stop bands (for all wave-polarizations) in the
frequency range $[16,21]$ Hertz (resp. $[6,11]$ Hertz) for $7.4$-meter (resp.
$0.74$-meter) diameter iron spheres with a $10$-meter (resp. $1$-meter)
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 and that rotational modes are totally overwritten by
bending modes. Regarding seismic applications, we further consider soil as a
bulk medium, in which case the relative bandwidth of 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 $49$ Hertz (relative stop band of $143$
percent) for iron spheres $0.74$-meter in diameter that are 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,
the foundations of large civil infrastructures.
