# Complex-Eigenfrequency Band Structure of Viscoelastic Phononic Crystals
## 3 Results
### Sonic Crystal of Rigid Rods in Air  
As a first example, we consider a two-dimensional sonic crystal of rigid rods in air, as was for instance considered experimentally by [PERSON] for cylindrical rods [20; 21] and by [PERSON] et al. for triangular rods [22]. The solid rods are supposed to be sufficiently heavy and stiff that only negligible elastic waves can be excited inside them for sound waves incident from air. As a result, the unit-cell of the crystal depicted in Figure 1a (square lattice) or Figure 1c (hexagonal lattice) is composed of a hollow air square or hexagon. With only a single fluid constituent, the theory of Section 2.3 can be applied. The viscosity of air is conventionally 0.0183 mPa.s at a temperature of 20 \({}^{\circ}\)C, as reported in Table 1. Combined with an elastic modulus \(B=142\) kPa, one would expect theoretically for longitudinal plane waves a quality factor \(Q=\frac{B}{\omega\eta}=12\),350 at a frequency of 100 kHz [3]. For the \(m\)-th band of the crystal, the quality factor is estimated from the complex eigenfrequencies as  
\[Q_{m}(\mathbf{k})=\frac{\text{Re}[\omega_{m}(\mathbf{k})]}{2\text{Im}[\omega_{ m}(\mathbf{k})]}. \tag{18}\]  
Figure 1 shows the complex-eigenfrequency band structures in the square and the hexagonal lattice cases, for lattice constant \(a=1\) mm. Also plotted are the empty-lattice models obtained when there is no rod in the unit-cell. Though the band structures are very different if the rod is present or  
epoxy matrix are an archetypal example, in both 2D [23] and 3D [24; 25]. With lattice constants of the order of millimeters, attenuation was not observed to be problematic for experiments, though epoxy materials are presumably viscoelastic. The viscosity of epoxy materials is quite variable, depending on the composition and on the processes used. We consider in Table 1 rather arbitrary values for the viscosity tensor of epoxy, for illustration purposes. Here, the viscosity of steel is neglected with respect to that of epoxy. For the 2D hexagonal lattice phononic crystal of cylindrical steel rods in an epoxy matrix depicted in Figure 2a, the complex-eigenfrequency band structure shown in Figure 2b as a large Bragg band gap [23] and the quality factor \(Q\) decreases again hyperbolically with increasing frequency.  
If the roles of steel and epoxy are reversed, for the case of epoxy rods in a steel matrix depicted in Figure 2c, then it is known that Bragg band gaps are not favored anymore but that local resonances are induced [26; 27; 28]. In the complex-eigenfrequency band structure shown in Figure 2d, such a local resonance appears for instance around a frequency of 1.25 MHz. It can be observed that the flatter bands have a much smaller quality factor compared to steeper bands. Actually, and despite appearances, this phenomenon is not related with band steepness but with the spatial distribution of Bloch waves within the unit-cell, as discussed in Section 4.  
Figure 2: Two-dimensional phononic crystals composed of epoxy and steel. The lattice is hexagonal, the lattice constant is \(a=1\) mm and \(d/a=0.7\). Only epoxy is considered viscoelastic while viscous loss in steel is neglected. (**a**) Unit cell for epoxy matrix and cylindrical steel rods, and (**b**) the corresponding complex-eigenfrequency band structure. The large complete band gap is of the Bragg type. (**c**) Unit cell for steel matrix and cylindrical epoxy rods, and (**d**) the corresponding complex-eigenfrequency band structure. The color scale of the band diagrams correspond to the quality factor \(Q\) of each Bloch eigenstate.