Analytic model of elastic metamaterials with local resonances

Abstract

A unified analytic model for effective mass density, effective bulk modulus, and effective shear modulus is presented for elastic metamaterials composed of coated spheres embedded in a host matrix. The effective material properties are derived directly from the averages of local momentum, stress, and strain defined in a single doubly coated sphere. It is shown that the effective material parameters predicted by the proposed model are in excellent agreements with the coherent-potential approximation results at low filling fractions where the anisotropy of periodic structures can be neglected for elastic waves. The advantage of the proposed method is that it can reveal clearly the physical mechanism for negative effective material parameters induced by the resonant effect. It is found that negative effective mass density is induced by negative total momentum of the composite for a positive momentum excitation. Negative effective bulk modulus appears for composites with an increasing ͑decreasing͒ total volume under a compressive ͑tensile͒ stress. Negative effective shear modulus describes composites with axisymmetric deformation under an opposite axisymmetric loading. Numerical examples are also given to illustrate these mechanisms. These findings may be useful in design of elastic metamaterials.

Negative refraction of elastic waves at the deep-subwavelength scale in a single-phase metamaterial

“…For isotropic EMMs, the three relevant effective properties are bulk modulus, shear modulus and mass density, the negative effective elastic parameters can be realized by promoting monopolar, dipolar and quadrupolar resonances with resonant microstructures into the building blocks of an EMM 27 , with the ultimate goal of attaining a negative index of refraction 28,29 . In these cases, the EMM can function as a left-handed medium, that is, S .…”

The band gap variation of a two dimensional binary locally resonant structure in thermal environment

“…According to the stress-strain relations from Hooke's law, the stress expression can be obtained from equation (2). The in-plane stress tensor is expressed in equation (4): 22…”

“…The unique characteristic of band gap in periodic structures or phononic crystals has attracted great attention, [1][2][3][4][5] because it provides a new solution for reducing vibration and noise in machinery, aerospace, architecture, transportation and other fields. Since the locally resonant phononic crystal (in short PC) was proposed in 1993, the band gap frequencies had reduced at least one or two orders of magnitude lower than that produced by the Bragg scattering mechanism.…”

Projeto e modelagem de metamateriais acústicos e elásticos por ressonâncias Mie.

“…Nos sistemas "solído-base", as inclusões esféricas podem ser fixadas na matriz hospedeira facilitando assim sua implementação no laboratório. A ressonância dipolar pode ser interpretada como uma oscilação fora de fase da esfera compósita em relação a um campo de força direcional, em que o núcleo de ouro fornece a massa pesada e a borracha D fornece a mola [17,48,49]. A seguir analisaremos um novo sistema "sólido-base" que exiba dupla ressonância numa faixa de frequências ampla.…”

“…Para entender o significado físico da aparição das duas regiões de frequência onde e  tornou-se negativo, nos explicamos a segui: A banda na região de baixa frequência acontece devido à oscilação fora de fase onde o núcleo da esfera do compósito atua como a massa pesada e a casca esférica de borracha M atua como a mola do sistema oscilante, na segunda banda, o núcleo permanece fixo e a casca esférica de borracha sofre um deslocamento na direção contraria ao movimento total da esfera do compósito [17,48,49].…”