Resonant and Non-Local Properties of Phononic Metasolids

We derive a general theory of effective properties in metasolids based on phononic crystals with low frequency resonances. We demonstrate that in general these structures need to be described by means of a frequency-dependent and non-local anisotropic mass density, stiffness tensor and a third- rank coupling tensor, which shows that they behave like a non-local Willis medium. The effect of non-locality and coupling tensor manifest themselves for some particular resonances whereas they become negligible for other resonances. Considering the example of a two-dimensional phononic crystal, consisting of triangular arrangements of cylindrical shells in an elastic matrix, we show that its mass density tensor is strongly resonant and anisotropic presenting both positive and negative divergent values, while becoming scalar in the quasi-static limit. Moreover, it is found that the negative value of transverse component of the mass density is induced by a dipolar resonance, while that of the vertical component is induced by a monopolar one. Finally, the dispersion relation obtained by the effective parameters of the crystal is compared with the band structure, showing a good agreement for the low-wave number region, although the non-local effects are important given the existence of some resonant values of the wave number.