Uncovering and Experimental Realization of Multimodal 3D Topological Metamaterials for Low-Frequency and Multiband Elastic Wave Control

Abstract

Topological mechanical metamaterials unlock confined and robust elastic wave control. Recent breakthroughs have precipitated the development of 3D topological metamaterials, which facilitate extraordinary wave manipulation along 2D planar and layer-dependent waveguides. The 3D topological metamaterials studied thus far are constrained to function in single-frequency bandwidths that are typically in a high-frequency regime, and a comprehensive experimental investigation remains elusive. In this paper, these research gaps are addressed and the state of the art is advanced through the synthesis and experimental realization of a 3D topological metamaterial that exploits multimodal local resonance to enable low-frequency elastic wave control over multiple distinct frequency bands. The proposed metamaterial is geometrically configured to create multimodal local resonators whose frequency characteristics govern the emergence of four unique low-frequency topological states. Numerical simulations uncover how these topological states can be employed to achieve polarization-, frequency-, and layer-dependent wave manipulation in 3D structures. An experimental study results in the attainment of complete wave fields that illustrate 2D topological waveguides and multi-polarized wave control in a physical testbed. The outcomes from this work provide insight that will aid future research on 3D topological mechanical metamaterials and reveal the applicability of the proposed metamaterial for wave control applications.

Keywords: elastic metamaterial; multiband waveguides; resonance; topological materials; wave control.

Figure 1

a) A schematic of the 3D topological metamaterial. b) Isometric and top views of the metamaterial unit cell. c) The band structure for the Type 0 lattice (α = 0). The four Dirac nodal line degeneracies are indicated by the dotted red boxes. The colorbar indicates the mode polarization quantified by the parameter Π. d) The band structure for the Type A/B (α = −0.11/0.11) lattices. The band structures for Type A and Type B lattices are identical and superimposed. The three split Dirac degeneracies are marked by gray shading. e) The mode shapes (taken along K‐H) for the bands that border the four (D1, D2, D3, and D4) topological bandgaps in the |α|= 0.11 case, illustrating multimodal resonance.

Figure 2

Parameter study illustrating the effect of the a) mass height h _m and b) spring ligament width w _l1 on the Dirac nodal line frequency f _d , which is taken at the midpoint between K and H for each D1‐D4 (f _d−D1 , f _d−D2 , f _d−D3 , and f _d−D4 ). All presented values are for the Type 0 lattice configuration (α = 0). The insets show the unit cell geometries for the minimum and maximum specified values of h _m and w _l1. The vertical dashed black lines indicate the specified h _m and w _l1 for all presented results in this paper. The dashed red lines indicate the minimum frequency of the ligament/rod modes.

Figure 3

a) Schematic of an eight‐unit supercell with a Type I interface indicated by the blue planar surface. b) The reciprocal space, with one‐half of the surface Brillouin zone projected onto the k _y−s − k _z−s plane outlined in light blue. c) The band diagram for the supercell presented in (a). The red bands (Λ _i  ≈ 1) are interface modes and the black bands (Λ _i  ≈ 0) are bulk modes. Representative mode shapes for the hybrid torsional (purple star), in‐plane torsional (purple circle), and in‐plane translational (purple square) topological interface states are shown at the bottom. d) The schematic and band diagram for a supercell comprised of eight Type B unit cells. The red bands (Λ _b  ≈ 1) are boundary modes and the black bands (Λ _b  ≈ 0) are bulk modes. Representative mode shapes for the topological boundary states are shown at the bottom.

Figure 4

a) Schematic of a full‐scale 3D metastructure constructed from an 8 × 8 × 6 pattern of the metamaterial unit cell (left) and illustrations of V‐shaped (middle) and Z‐shaped (right) waveguides. The distribution of Type A and Type B unit cells is denoted by the letters “A” and “B.” b) Steady‐state displacement fields illustrating waveguides for all‐layer (L1‐L6) input excitations of 1.3 kHz (hybrid torsional), 3.6 kHz (in‐plane torsional), and 4.9 kHz (in‐plane translational). c) Wave attenuation when an out‐of‐plane excitation is used in the frequency ranges of the in‐plane topological states. d) Steady‐state displacement fields illustrating layer‐locked waveguiding for two‐layer (L1 and L4) input excitations of 1.4 kHz (hybrid torsional), 3.6 kHz (in‐plane torsional), and 4.9 kHz (in‐plane translational). For a two‐layer input of 1.3 kHz, the hybrid torsional state exhibits wave transmission across all six layers.

Figure 5

a) The experimental testbed with an inset showing detail for a unit cell. b) A topological boundary state with hybrid torsional polarization that is found in the band structure of a four‐unit supercell. c) Experimentally measured out‐of‐plane velocity field (left) and finite element simulated displacement field (right) for the 3D metastructure obtained at f _m1 = 1.3 kHz. For clarity, both the full‐scale and layer views of the simulated displacement field are shown. The schematic of the 3D metastructure testbed is given in the top right, where the blue shading represents the V‐shaped waveguide. d) The experimentally measured out‐of‐plane velocity (v _op ) for Point A (solid lines) and Point B (dashed lines) on each of the four layers. The frequency range for effective waveguiding is marked by the gray shading. e) Experimentally measured out‐of‐plane velocity fields for L4 that illustrate the topological waveguide across a wide frequency range. A bulk response is demonstrated at 1.54 kHz.

Figure 6

a) (top) A schematic of the 3D metastructure testbed where the red shading represents the path of a V‐shaped waveguide. (bottom) A topological boundary state with an in‐plane torsional polarization that is found in the band structure of a four‐unit supercell. b) Experimentally measured in‐plane velocity field and finite element simulated displacement field for L4 of the 3D metastructure obtained at f _m2 = 3.7 kHz. c) The experimentally measured in‐plane velocity (v _ip ) for a Point Inside and a Point Outside the waveguide on L4. The frequency range for effective waveguiding is marked by the gray shading. d) (top) A schematic of the 3D metastructure testbed where the blue shading represents the path of a V‐shaped waveguide. (bottom) A topological boundary state with an in‐plane translational polarization that is found in the band structure of a four‐unit supercell. e) Experimentally measured in‐plane velocity field and finite element simulated displacement field for L4 of the 3D metastructure obtained at f _m3 = 4.4 kHz. f) The experimentally measured in‐plane velocity (v _ip ) for a Point Inside and a Point Outside the waveguide on L4. The frequency range for effective waveguiding is marked by the gray shading. See Section S9 (Supporting Information) for the transmission ratio plots that accompany c) and f). g) Experimentally measured in‐plane velocity fields for L4 that illustrate the in‐plane topological waveguides across a wide frequency range. A bulk response is demonstrated at 4.12 kHz.