Intermodal and Subwavelength Energy Trapping in Nonlinear Metamaterial Waveguides

In this work, we experimentally demonstrate the phenomenon of nonlinearity-activated intermodal tunneling in a periodic elastic metamaterial waveguide with internal resonators and we show how this effect can be exploited to achieve conspicuous energy localization and trapping. The architecture of the waveguide is deliberately designed to promote tunneling from flexurally dominated to axially dominated modes, in order to accentuate the functional complementarity that can be harnessed during tunneling. 3D laser vibrometry at different scales of spatial refinement is employed to capture global and local in-plane features of the wavefield. The measured response naturally yields an experimental reconstruction of the band diagram of the waveguide and reveals unequivocally the spectral signature of the high-frequency modes that are activated by tunneling. Finally, a detailed scan of selected cells highlights a strong and persistent axial activation of the resonators, which displays subwavelength deformation features that are unattainable, for the axial mode, by exciting at the same frequency in a linear regime. This result demonstrates the viability of tuning strategies based on nonlinearity and paves the way for the design of metastructures with enhanced energy-trapping and harvesting capabilities.

Figure 1

(a) Waveguide specimen with detail of a unit cell with a steel-core resonator. (b) Experiment setup. (c) Detail of the scanned surface. (d) Shaker position for flexural excitation.

Figure 2

Band diagram of the waveguide with mode shapes shown for three representative spectral points. $\xi$ is the nondimensional wavenumber ($\xi =ka$, where $k$ is the wavenumber and $a$ is the length of the unit cell).

Figure 3

Experimentally reconstructed spectral response and comparison to FE data. (a) Lateral velocity field. (b) Nondispersive region and (c) steering region of the axial mode.

Figure 4

(a) Frequency spectrum of a tone-burst excitation. (b) Spectral representation of the response collected at a single scan point. (c) Ratio between the amplitude of the axial second harmonic and the amplitude of the fundamental flexural harmonic for four different amplitudes (scaled by the maximum value) of excitation. (d),(e) Spectral representation (via $2\mathrm{D}$ DFT) of the lateral and axial response, respectively, sampled along the waveguide axis.

Figure 5

Schematic illustration of quadrant shift due to aliasing during intermodal tunneling.

Figure 6

Experimentally reconstructed internal unit-cell response: snapshots of local wavefield details (spanning three cells) at four successive time instants (normalized as ${\overline{t}}_i=t_i/T$, where $T=0.5\mathrm{ms}$ and $i=1,2,3,4$). (a) Fundamental flexural wavefield. (b) Filtered wavefield at the second harmonic revealing nonlinear activation of axial resonant mechanisms and energy trapping.