# Multi-parametric sensitivity analysis of the band structure for tetrachiral acoustic metamaterials

## TL;DR

This paper provides an analytical framework for understanding and designing the band structure of tetrachiral acoustic metamaterials, enabling precise control over wave propagation and band gap creation through sensitivity analysis and perturbation techniques.

## Contribution

It introduces a parametric description of the band structure and applies multiparametric perturbation methods to analytically approximate dispersion functions and design band gaps.

## Key findings

- Analytical conditions for full band gaps in low-frequency range.
- High-accuracy parametric approximations of the band structure.
- Design of target stop bands using inverse spectral problems.

## Abstract

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.