Discretization Approach for the Homogenization of Three-Dimensional Solid-Solid Phononic Crystals in the Quasi-Static Limit: Density and Elastic Moduli

Abstract

With the application of a homogenization theory, based on the Fourier formalism (which provides efficient and exact formulas by which to determine all the components of the effective stiffness and mass density tensors, valid in the regime of large wavelengths), a new approach to calculate the effective quasi-static response in three-dimensional solid-solid phononic crystals is reported. The formulas derived in this work for calculating the effective elastic parameters show a dependence, in terms of summations over the vectors, of the reciprocal lattice by the discretization of the volume of the inclusion in small parts (e.g., small cubes), to obtain a system of equations from which we define the effective response. In particular, we present the numerical results calculated for several cubic lattices with solid constituents and different shapes of inclusions in the unit cell versus the filling fraction, as well as for fixed values of it. By this means, we analyzed the effect of the type of Bravais lattice of the materials, and the geometry of the inclusions that constitute the three-dimensional phononic array, on the resulting effective anisotropy. Finally, our theory confirms other well-known results with previous homogenization theories as a particular case study. In this regard, the examples and results shown here can be useful for the design of metamaterials with predetermined elastic properties.

Phononic Band Structure by Calculating Effective Parameters of One-Dimensional Metamaterials

“…The numerical approximation used in this study is based on the recent theoretical homogenization technique described in [45]. In that work, the effective properties of the artificial composite material are derived from the fact that, as it is a periodic structure, both its material parameters (e.g., moduli of elasticity and density) as well as their reciprocals that are in function of the position, can be expanded into Fourier series.…”

“…According to this approximation (see [45] and analytical deduction therein), the general equation for calculating the effective tensors of the effective material response as a function of the frequency and the wave vector (dynamic case), is given by the expression:…”

“…One of the common conditions to conform resonant metamaterials (i.e., negative simultaneous density and shear modulus) is the high contrast between the elastic properties of their constituents [39][40][41][42][43][44]. Considering the above, in this article we will analyze from a homogenization method that allows to study the material effective response for any form of the inclusions within the unit cell of a 3D phononic crystal and whose principle lies in the discretization of the inclusion [45], the frequency dependence of the effective tensors of moduli of elasticity and density for a transverse polarization of a 1D phononic crystal. For a filling fraction established for the inclusion, it is shown that the effective dynamic parameters can reproduce the exact dispersion relations for the acoustic modes that propagate along the periodicity direction of the crystal.…”