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 parameters of arrangement of inclusions or resonators in thin plates as a function of the filling fraction of the inclusions, their physical properties and the frequency. It is shown that positive or negative effective elastic parameters are possible depending on the symmetry of the resonance but, unlike it happens for bulk elastic waves, the responsible for the negative mass density behaviour is the monopolar term, while the negative Young's modulus and Poisson's ratio is due to the combination of monopolar and quadrupolar resonances, showing also that, at least for the first order in the scattering coefficients, the dipolar resonance plays no role in the description of the effective medium. Several examples are given for both non-resonant and resonant effective parameters and the results are verified by multiple scattering theory.

References in corpus (6)

• All-Dielectric Rod-Type Metamaterials at Optical Frequencies
• Sound focusing by gradient index sonic lenses
• Sonic Gradient Index Lens for Aqueous Applications
• Acoustic resonances in two dimensional radial sonic crystals shells
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Cited by in corpus (11)

• Gradient index phononic crystals and metamaterials
• The Inverse Grating Problem: Efficient Design of Anomalous Flexural Wave Reflectors and Refractors
• Resonant and Non-Local Properties of Phononic Metasolids
• Dipolar localization of waves in twisted phononic crystal plates
• Omnidirectional refractive devices for flexural waves based on graded phononic crystals
• Dynamic homogenization of composite and locally resonant flexural systems
• Scattering of flexural waves from an $N$-beam resonator in a thin plate
• Linking scalar elastodynamics and non-Hermitian quantum mechanics
• Flexural Mie Resonances: Localized Surface Platonic Modes
• Metamaterials: $\text{supra}$-classical dynamic homogenization
• Fundamental principles for generalized Willis metamaterials