Resonant and nonlocal 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 nonlocal anisotropic mass density, stiffness tensor and a third-rank coupling tensor, which shows that they behave like a nonlocal Willis medium. The effect of nonlocality 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 quasistatic 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 good agreement for the low-wave-number region, although the nonlocal effects are important given the existence of some resonant values of the wave number.

Figure 1

Phononic crystal studied in the present work. The system consists of a triangular arrangement of coated cylinders in an epoxy background. The cylinders consist of a lead core of radius $r_a$ and a rubber shell of radius $r_s$ (see text for numerical values).

Figure 2

Effective mass density tensor for the proposed phononic crystal. Both the transversal component (blue continuous line) and the $z$ component (green dashed line) present a local resonance. In the vicinity of this resonance both components of the mass density become negative. The red dot indicates the frequency and density of the example used in Fig. 5 (see text for further details).

Figure 3

Resonant modes inducing an effective negative mass density. Upper panels for ${\omega }_{k}a/2\pi c_t=0.0072$, corresponding to a resonance in ${\rho }_z$, and lower panels for ${\omega }_{k}a/2\pi c_t=0.0194$, corresponding to a resonance in ${\rho }_{\perp }$ (see text for further discussion).

Figure 4

Left panel: Dispersion relation of the phononic crystal along the $\mathrm{\Gamma }X$ direction (black lines) compared with those obtained from the effective local constitutive parameters (green crosses and blue and red dots). Right panel: Dispersion relation of the phononic crystal along the $\mathrm{\Gamma }A$ direction (black lines) compared with those obtained from the effective local constitutive parameters (green crosses and blue dots).

Figure 5

Nonlocal constitutive parameters related with the propagation of the $z$ mode in the phononic crystal. Upper panel: $z$ component of the mass density. Middle panel: $C_{44}$ and $C_{33}$ components of the stiffness tensor. Lower panel: $S_{53}$ and $S_{33}$ components of the coupling field (see text for details).