High-frequency parametric approximation of the Floquet-Bloch spectrum for anti-tetrachiral materials

Abstract

The class of anti-tetrachiral cellular materials is phenomenologically characterized by a strong auxeticity of the elastic macroscopic response. The auxetic behavior is activated by rolling-up deformation mechanisms developed by the material microstructure, composed by a periodic pattern of stiff rings connected by flexible ligaments. A linear beam lattice model is formulated to describe the free dynamic response of the periodic cell, in the absence of a soft matrix. After a static condensation of the passive degrees-of-freedom, a general procedure is applied to analyze the wave propagation in the low-dimensional space of the active degrees-of-freedom. The exact dispersion functions are compared with explicit – although approximate – dispersion relations, obtained from asymptotic perturbation solutions of the eigenproblem governing the Floquet–Bloch theory. A general hierarchical scheme is outlined to formulate and solve the perturbation equations, taking into account the dimension of the perturbation vector. Original recursive formulas are presented to achieve any desired order of asymptotic approximation. For the anti-tetrachiral material, the fourth-order asymptotic solutions are found to approximate the dispersion curves with fine agreement over wide regions of the parameter space. The asymptotic eigensolutions allow an accurate sensitivity analysis of the material spectrum under variation of the key physical parameters, including the cell aspect ratio, the ligament slenderness and the spatial ring density. Finally, the explicit dependence of the dispersion functions on the mechanical parameters may facilitate the custom design of specific spectral properties, such as the wave velocities and band gap amplitudes

Acoustic wave polarization and energy flow in periodic beam lattice materials

“…where ∂ λ and ∂ βi stand for the λ-derivative and the β i -derivative, respectively, with β i component of b = (β 1 , β 2 ) and i = 1, 2. An analytical formula for the asymptotic approximation of the derivatives in the expression (49) can be found in [57]. An alternative method to determine the group velocity can be founded on the β i -derivative of the condensed equations of motion (3).…”

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

“…The wave equation (6) can be tackled by imposing the harmonic mono-frequent solution q a = ψ a exp(iωτ), where ω = Ω/Ω c and Ω are the unknown nondimensional and dimensional wave frequency, respectively. Therefore, eliminating the dependence on time, a linear eigenproblem can be obtained in the standard form…”

“…Perturbation methods can represent an efficient alternative tool, suited to determine explicit -though asymptotically approximate -analytical expressions of the dispersion relation for periodic structures [24][25][26][27], as well as for their equivalent homogenized continua [28]. In [6], a multi-parameter perturbation strategy has been outlined to build up asymptotic approximations for the dispersion functions of Lagrangian lattice models. The strategy is based on including both the wavenumbers and the mechanical parameters in the small amplitude perturbation vector µ = (p , k ), spanning a small-radius multidimensional hyper-sphere centered at a suited reference point µ • of the multi-parameter space.…”

Bloch wave filtering in tetrachiral materials via mechanical tuning

“…In this respect, the periodic materials with a chiral or antichiral microstructure of the elementary cell [18], [19], [20], consisting of stiff disks or rings, connected by a variable number of flexible ligaments, are particularly attractive for their potential as acoustic waveguides or phononic filters. In the current literature dealing with this material class, the pass and stop bands characterizing the band structures have been determined by solving the dispersion problem related to low-dimensional lagrangian models [21], [22], [23], [24], [25], [26] high-fidelity micromechanical formulations accounting for the material heterogeneity at the microscale [11], [12], [14], [27] and equivalent local and non-local homogenized continua [9], [24], [25], [26]. The underlying idea is that, within certain physically admissible ranges, the geometric and mechanical parameters can be intended as freely tunable variables for customizing the acoustic dispersion properties of the material.…”

“…horizontally and vertically propagating waves, respectively.The conditions (4) can be introduced in the quasi-static equation(3)to reduce the number of independent passive displacements[23]. Therefore, the linear quasi-static law between the passive displacements or forces and the active degrees-offreedom.…”