  Tetrachiral materials are characterized by a cellular microstructure made by
a periodic pattern of stiff rings and flexible ligaments. Their mechanical
behaviour can be described by a planar lattice of rigid massive bodies and
elastic massless beams. The periodic cell dynamics is governed by a monoatomic
structural model, conveniently reduced to the only active degrees-of-freedom.
The paper presents an explicit parametric description of the band structure
governing the free propagation of elastic waves. By virtue of multiparametric
perturbation techniques, sensitivity analyses are performed to achieve
analytical asymptotic approximation of the dispersion functions. The parametric
conditions for the existence of full band gaps in the low-frequency range are
established. Furthermore, the band gap amplitude is analytically assessed in
the admissible parameter range. In tetrachiral acoustic metamaterials, stop
bands can be opened by the introduction of intra-ring resonators. Perturbation
methods can efficiently deal with the consequent enlargement of the mechanical
parameter space. Indeed high-accuracy parametric approximations are achieved
for the band structure, enriched by the new optical branches related to the
resonator frequencies. In particular, target stop bands in the metamaterial
spectrum are analytically designed through the asymptotic solution of inverse
spectral problems.
