Dynamic mass density of resonant metamaterials with homogeneous inclusions

The occurrence of a negative dynamic mass density is a striking property of metamaterials. It appears when an inner local resonance is present. Results coming from an asymptotic theory are recalled briefly, showing the scaling of physical properties leading to inner resonance in elastic composites containing homogeneous soft inclusions, with negligible scattering of waves traveling through the matrix. This appears for a large contrast of elastic properties between matrix and inclusion. The frequency-dependent dynamic mass density depends on the resonance frequencies of the inner inclusions and on their related participation factors. Having solved the dynamic elasticity problem, these physical quantities are provided in the case of homogeneous cylindrical and spherical inclusions. It is shown that numerous resonance frequencies do not contribute to the dynamic mass density or have small participation factors, which simplifies significantly the physics involved in the concerned inner resonance phenomena. Finally, non-dimensional resonance frequencies and participation factors are given for both cases of inclusions as functions of the Poisson's ratio, defining completely the dynamic mass density.