Effective medium theory for elastic metamaterials in thin elastic plates

Abstract

An effective medium theory for resonant and non-resonant metamaterials for flexural waves in thin plates is presented. The theory provides closed-form expressions for the effective mass density, rigidity and Poisson's ratio of arrangements of isotropic scatterers in thin plates, valid for low frequencies and moderate filling fractions. It is found that the effective Young's modulus and Poisson's ratio are induced by a combination of the monopolar and quadrupolar scattering coefficient, as it happens for bulk elastic waves, while the effective mass density is induced by the monopolar coefficient, contrarily as it happens for bulk elastic waves, where the effective mass density is induced by the dipolar coefficient. It is shown that resonant positive or negative effective elastic parameters are possible, being therefore the monopolar resonance the responsible of creating an effective medium with negative mass density. Several examples are given for both non-resonant and resonant effective parameters and the results are numerically verified by multiple scattering theory and finite element analysis.

Omnidirectional refractive devices for flexural waves based on graded phononic crystals

“…It is known that, for wavelengths larger than the lattice constant a, this cluster behaves as an effective medium with certain effective parameters, which depend on the physical properties of the inclusions and their size. In reference [21] it was demonstrated that the effective parameters for this arrangement of inclusions is obtained by means of the following equations…”

“…In this approach, the device is consist of a circular cluster of inclusions arranged in an ordered lattice. This cluster behaves, in the low frequency limit, as a homogeneous device whose elastic parameters depend on the physical nature of the inclusions and their size [21]. Then, if the size of the inclusions is changed according to their position, following a specific law, the refractive index of the material can be changed accordingly.…”

Aerogel as a Soft Acoustic Metamaterial for Airborne Sound

“…Therefore, a more useful quantity is the spatial average of the displacement, which can be obtained from Eq. (16) such that…”

“…One of the fundamental aspects that gives a metamaterial its exotic macroscopic properties is the dynamics of the microstructure and the resulting "hidden" degrees of freedom that it enables [15], which has been explored recently for elastic and flexural metamaterial components [16,17]. These hidden degrees of freedom are particularly important because the effective macroscopic properties of an acoustic metamaterial can be significantly different than those of the constituent microstructural elements.…”

Focusing of the lowest-order antisymmetric Lamb mode behind a gradient-index acoustic metalens with local resonators

“…However, none of the proposed designs has allowed us to overcome the diffraction limit so far [26][27][28][29][30][31][32][33][34][35][36] because of total internal reflection of the waves with high k vectors which hinders the transmission to the background for external focusing. It was further proposed to use metamaterials featuring extraordinary refractive properties instead of photonic crystals [37][38][39][40][41], and the resolution beyond the diffraction limit for visible light was actually demonstrated analytically [37]. However, such a success has not been obtained with elastic waves in solids so far [38][39][40][41], nor has it been investigated how large should be the acoustical refractive index (i.e., ratio of the k vector in the lens to the k vector in the background) to attain this objective.…”