Polarization bandgaps and fluid-like elasticity in fully solid elastic metamaterials

Elastic waves exhibit rich polarization characteristics absent in acoustic and electromagnetic waves. By designing a solid elastic metamaterial based on three-dimensional anisotropic locally resonant units, here we experimentally demonstrate polarization bandgaps together with exotic properties such as ‘fluid-like' elasticity. We construct elastic rods with unusual vibrational properties, which we denote as ‘meta-rods'. By measuring the vibrational responses under flexural, longitudinal and torsional excitations, we find that each vibration mode can be selectively suppressed. In particular, we observe in a finite frequency regime that all flexural vibrations are forbidden, whereas longitudinal vibration is allowed—a unique property of fluids. In another case, the torsional vibration can be suppressed significantly. The experimental results are well interpreted by band structure analysis, as well as effective media with indefinite mass density and negative moment of inertia. Our work opens an approach to efficiently separate and control elastic waves of different polarizations in fully solid structures.

3 / TM(x, y) are in red / blue (green), respectively. Fluid-like property is found for waves that propagate in z direction, which is shaded dark grey (~1.1-1.7 kHz). Light grey region (~1.7-2.9 kHz) is partially fluid-like: zpolarized shear waves cannot propagate, but both longitudinal and y-polarized shear waves are allowed. Meta-rods with circular cross section are shown in a, b; and meta-rods with rectangular cross section (aspect ratio 2:1) are shown in c, d. In all panels, red markers delineate branches associated with longitudinal modes;

Supplementary
blue/green markers plot branches associated with flexural modes, and orange/cyan markers show torsional branches. Fluid-like elasticity can be found in both cases in z-stack meta-rods (grey shaded regions in a, c).
The overall behaviors of the circular meta-rods are almost identical to the square meta-rods (the case studied in the main text). Rectangular meta-rods have additional anisotropy in their cross sections, consequently the two flexural branches split and have different propagation speed (enclosed by dashed ovals).

Supplementary Information
Polarization Bandgaps and Fluid-like Elasticity in Fully Solid Elastic Metamaterials 5 Supplementary Figure 7 For a rod with rectangular cross-section, the two flexural modes are no longer degenerate, but have different dispersions. If the rod the bending along the thicker direction (blue inset), the flexural modulus in effect is larger, thereby the wave speed is generally higher (blue curve). If the rod the bending along the thinner direction (red, inset), a smaller flexural modulus leads to lower wave speed (red curve). This observation will lead to interesting consequences for meta-rods as well (see Supplementary Fig. 6c, d).
Supplementary Figure 8 Tuning the eigenfrequencies by material parameters. In a, the mass density of the cylindrical core is altered, whereas the silicone layer and epoxy maintain their properties. It is seen that all eigenfrequencies decrease as the core's mass density increases. In b, the Young's modulus of the silicone is

Supplementary Information
Polarization Bandgaps and Fluid-like Elasticity in Fully Solid Elastic Metamaterials 6 varied. Clearly, the harder (larger Young's modulus) the layer is, the higher the eigenfrequencies. Here, steel and epoxy are used as the materials of the core and cladding. In c, the mass density of the cladding (or background) is adjusted, whereas steel and silicone are used for the core and coating layer. Eigenfrequencies drop as the mass density increases.

Supplementary Note 4 Torsional response of sample-
Torsional vibration can also be excited in sample-by the setup shown in Fig. 3c (main text). From the unit cell's orientation, it is easy to see that the rotational mode about z-axis, i. e., RM(z), is excited.
Mode profile of RM(z) is shown in Fig. 1c (main text). Naturally, RM(z) couples with the rod's torsional vibration, giving rise to a polaritonic dispersion, together with a bandgap. The calculated band structure of sample-is shown again in Supplementary Fig. 3a, with the torsional branch highlighted in cyan, and the bandgap shaded cyan. This bandgap is confirmed by finite element simulation of the response function, which found a response minimum near 0.75 kHz, as shown in Supplementary Fig. 3b. We have also calculated , the effective moment of inertia per unit cell about z-axis. We found that < 0 in 0.7-0.8 kHz (dashed curves in Supplementary Fig. 3b, right axis). interface. This is particularly likely when the interface is curvilinear, which is exactly the case for RM(z). Consequently, RM(z) may not be effectively excited even with rotational actuation, which results in the unsuccessful experimental attempts.
In contrast, for RM(x) and RM(y), the coupling strength must be larger, as indicated by the larger bandgap size (>200 Hz, orange shaded region in Fig. 2b, main text). This owes to the better structural restriction: for the steel cylinder to rotate about x or y-axis, there shall always be a certain degree of compression/expansion in the silicone rubber when relative rotational motions occur, since the side and ends of the steel cylinder meet in right angles. Therefore, the requirement on interface bonding is also far less stringent.

Supplementary Note 5. Numerical study of a three-dimensional array
By repeating the same metamaterial unit cells in all three spatial directions, a three-dimensional (3D) lattice can be constructed. Its band structure is shown in Supplementary Fig. 4. Here, z is the direction parallel to the steel cylinder's axis; and x, y are in-plane with the cylinder's ends. Likewise, the lattice is isotropic in x and y, and is anisotropic in z.
We make the following observations in Supplementary Fig. 4 The unique elastic properties of the 3D metamaterial can also emerge from an effective medium.
Since all translational eigenmodes are clearly dipolar in their symmetry, they should dominantly induce anomaly in components of effective mass density tensor 3-5 . Due to the anisotropic unit structure, the effective mass density is also anisotropic. The effective mass density can be obtained from the Newton's second law 4,6 , i.e.
Here is the effective mass density in the i direction; is the effective net force exerted on the unit cell in the i direction; is the effective displacement of the unit cell in the i direction, the double over-dots indicating second order time derivatives; and is the lattice constant. and can both be obtained from the eigenfunctions.
We note that different eigenmodes should be chosen to obtain effective mass density in different directions. For waves propagating in direction, we have For waves propagating in direction, we have Here, are the components of the stress tensor, with , = , , .
The effective mass densities obtained by eigenmodes in are shown in Supplementary Fig. 5a. It is seen that in 1.1-1.7kHz, we have < 0, < 0, > 0.
In electromagnetism, anisotropic effective parameters with different signs along different spatial directions are dubbed "indefinite" 7 . We therefore use the name "indefinite effective mass density". For It is also seen that in 1.9-2.9 kHz, > 0, > 0, < 0.
Namely, shear waves are allowed, whereas longitudinal waves are not.
We have further calculated the transmission coefficients for all three types of waves using COMSOL Multiphysics. The system parameters used here are identical to those shown in "Methods" following the main text. Dissipation is also added to silicone rubber. The results are shown in Supplementary Fig. 5b. We put six unit cells in the direction of incidence, whereas full periodicity are used for the four boundaries parallel to incidence. Excellent agreement with band structure analysis and effective medium prediction is achieved.
Effective mass densities of waves that propagate along are shown in Supplementary Fig. 5c.
Therefore, longitudinal waves and y-polarized shear waves cannot propagate in this frequency regime.
Therefore, longitudinal and y-polarized shear waves are allowed, but z-polarized shear waves are forbidden. This regime is partially fluid-like. Likewise, simulated transmission coefficients confirm our analysis (Supplementary Fig. 5d).
In previous studies on acoustic metamaterials, the anisotropic mass density with both positive and negative components, i.e. indefinite mass density, have been shown to lead to hyperbolic dispersions [8][9][10] , similar to electromagnetic metamaterials 11 . Such unique hyperbolic dispersion leads to important applications such as hyperlenses 10,12,13 . In the context of elastic waves, however, the indefinite mass

Supplementary Note 6. Geometry of meta-rod's cross section
Here, we investigate how the cross sectional shape affects the wave properties of the meta-rod. We keep all material parameters the same as those in the main text. The steel cylinder, silicone layer, and lattice constant also retain their geometric values. First, we note that both flexural modulus and torsional rigidity depend on the rod's cross sectional shape 1 . Therefore changing the cross sectional geometry will affect both quantities. For a torsional wave, a different torsional rigidity merely induces a different wave speed. However, for flexural waves the situation is slightly more interesting.
If a rod has cross-sectional shape of a circle, then it is considered isotropic in its cross-sectional plane. Flexural waves with both polarizations must still have the same dispersion relation and the same wave speed (at a given frequency). Consequently, overall behaviors of circular meta-rods can be almost identical to the square meta-rods. This can be seen in Supplementary Fig. 6a, b. In which all polarization bandgaps and the fluid-like region (grey shaded in Supplementary Fig. 6a) can be easily identified for both z-stack (in which the steel cylinders' axes are aligned and are parallel to the metarod) and x-stack (in which the steel cylinders' axes are perpendicular to the meta-rod) cases. Note that z-stack meta-rod is homogeneous in xy-plane, therefore its two flexural branches are degenerate (marked by black dashed ovals in Supplementary Fig. 6a).
If additional anisotropy is introduced, for instance, the rod's cross-sectional shape is changed into a rectangle, then flexural waves will pick up different dispersion relations for different polarizations.
This is schematically explained in Supplementary Fig. 7 for a homogenous rectangular rod. If the rod is bending along the rectangle's long side (blue inset), the flexural modulus takes a larger value and the wave speed is generally higher (blue curve). In contrast, if the rod is bending along the rectangle's short side (red inset), the wave speed is lower (red curve). This observation also appears for meta-rods. mass density of the cylindrical core (silicone and epoxy kept unchanged), the Young's modulus of the elastic layer (steel core and epoxy unchanged), and the mass density of the cladding material (steel core and silicone unchanged). All geometric parameters are identical to the unit cell in the main text.
The results are shown in Supplementary Fig. 8. In short, increasing the mass density, or lowering Young's modulus leads to the reduction in the eigenfrequencies.
As a further demonstration, we numerically study the response functions of two specific examples using some widely available materials: granite coated by foam embedded in spruce, and bronze coated by latex rubber embedded in marble. We use the following material parameters ( is mass density,