Analytic model of elastic metamaterials with local resonances

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 (decreasing) total volume under a compressive (tensile) 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.