Soft 3D Metamaterial for Low-Frequency Elastic Waves

TL;DR

The study presents a fully soft 3D metamaterial with liquid-metal resonators embedded in a compliant SLA lattice, achieving a subwavelength band gap near $f\approx 200\,\mathrm{Hz}$ for low-frequency elastic waves. A hybrid workflow combining a lumped-element model and 3D-FEA with Floquet–Bloch analysis guides design (e.g., $a=16.5\,\mathrm{mm}$, $r=0.42$) to open the band gap around $[185,208]\,\mathrm{Hz}$, later validated by optical vibrometry and accelerometer measurements showing a complete gap $[200,340]\,\mathrm{Hz}$ (velocity) and $[210,330]\,\mathrm{Hz}$ (acceleration) with strong attenuation and deeply subwavelength propagation ($\lambda_y/a \approx 9.9$ at peak). The dense, low-viscosity Galinstan inclusions decouple flexural and torsional modes, enabling robust low-frequency attenuation with an effective density of $\rho_{\mathrm{eff}} \approx 0.53\,\mathrm{g/cm^3}$, outperforming common elastomers at half the weight. The approach offers scalable fabrication and tunable unit cells for applications in haptics, padding, and vibration isolation, with potential extensions to seismic regimes through further miniaturization or upscaling. A general principle emerges: combining materials in distinct physical states can create avoided crossings that open band gaps in soft metamaterials.

Abstract

Acoustic metamaterials offer exceptional control over wave propagation, but their potential remains unfulfilled due to fabrication constraints. Conventional processes yield mostly rigid, planar structures, whereas soft-matter alternatives have so far been confined to ultrasounds. This work overcomes prior limitations with a fully soft 3D metamaterial operating around 200Hz. The design combines a 3D-printed elastomer lattice with resonant inclusions of liquid metal, injected via a network of mesofluidic channels. Its dynamic response is derived from a hybrid strategy uniting a lumped-element model with finite element analysis. Simulations reveal how the dual-phase design decouples flexural and torsional modes, opening a subwavelength band gap for low-frequency elastic waves. Empirical validation is achieved via a custom camera-based vibrometer. Its high spatiotemporal resolution and full-field capabilities enable direct capture of local modes and evanescent waves underlying the band gap. Accelerometer data corroborate these findings and demonstrate greater attenuation than common silicone elastomers at only half of the density. By combining scalable fabrication, compliance, and operations at frequencies relevant to human tactile perception, this novel metamaterial paves the way for lightweight, high-performance cushioning and handles that protect users from harmful vibration exposure.

Figures (5)

• Figure 1: A) Cross-section of a unit cell with a resonant inclusion of liquid metal held by orthogonal soft rods. B) Schematic of our fully soft 3D metamaterial, restricted to a cubic lattice of $4^3$ unit cells for practicality. C) Slice revealing the U-shaped mesofluidic channels used for filling. Colored fluid is for visualization only. D) Comparison with state-of-the-art soft metamaterials for elastic or acoustic wave propagation across aqueous, silicone, and gel-like media. Our design operates at frequencies nearly two orders of magnitude lower. Characteristic size is defined as the lattice constant for periodic structures or as the microparticle diameter for disordered colloidal dispersions. E) Transmissibility measurements along the three axes of our metamaterial. Narrower violin plots indicate stronger attenuation. Band gaps are shown as shaded areas, with their overlap highlighted. Our metamaterial achieves greater attenuation than ordinary silicone, here taken as Ecoflex 00-10, at a fraction of the density. F) Potential use in haptics as a low-frequency vibration insulator, contextualized alongside existing quasi-static haptic metamaterials.
• Figure 2: A) Lumped 1D mass-spring model of a unit cell. B) Meshed 3D finite-element model of a unit cell. Specific boundaries are highlighted. C) Numerically computed band gap and density for $a=16.5mm$. Markers are discrete simulation steps. The dotted line shows the chosen design. D) Frequency-dependent effective lumped mass for lattice constants $a$ and a ratio $r=0.42$. The shaded region outlines a preliminary band gap estimate. E) Tetrahedral irreducible Brillouin zone. F) Resonance frequency of the lumped model as a function of $a$ and $r$. Shaded areas denote $r$ values excluded by fabrication constraints. The dotted curve shows the theoretical optimum $r_{\!o}$, from which the chosen design deviates to ensure manufacturability. G) Dispersion graph of the tuned lattice with $a=16.5mm$ and $r=0.42$. A complete band gap is bounded by flexural and torsional modes.
• Figure 3: A) Finite element analysis of local vibration modes framing the band gap in an infinite lattice. The deformed shapes, magnified for clarity, are depicted at wave vector $\boldsymbol{k}^\mathrm{X}$ for $\mathrm{M}_2$ and $\mathrm{M}_3$, with $\boldsymbol{k}^\mathrm{R}$ for the remaining modes. Normalized amplitudes of solid displacement and liquid acceleration are shown, with pressure isobars overlaid. B) High-speed imaging setup for full-field vibrometry. C) Bode diagram, averaged over ten standalone unit cells and fitted by a third-order model. The shaded area shows $\pm 1$ SD. Inset shows the Hankel singular value decomposition. D) Photograph of a standalone unit cell, about the size of a fingertip. E) Image-processing workflow for tracking the vertical motion of the liquid metal and outer shell centroids. F) Resulting empirical harmonic response, e.g. for a 220Hz input of standalone cell No. 3.
• Figure 4: A) Spatiotemporal evolution of the $y$-axis particle velocity shown at 0.5ms intervals, i.e. one in every two frames. Examples are given for 220Hz and 420Hz sine inputs, inside and above the band gap $\mathrm{BG}_y$, respectively. The top row reveals evanescent waves while the bottom row shows waves propagating throughout the metamaterial. B) Spatially averaged $y$-axis particle velocity in each layer $\mathrm{L}_j$ for 220Hz and 420Hz. C) Phase velocity derived by linear regression of the phase lag between layers. D) Optical vibrometry setup for capturing waves on the metamaterial outer shell. E) Photograph of the prototype made of $4^3$ unit cells. F) Empirical dispersion diagrams. Shaded areas denote hybridization band gaps. Dotted curves are second-order polynomial model of the propagating modes, fitted to the data with tangency enforced at the band gap boundaries.
• Figure 5: A) RMS particle velocity along the $y$-axis, normalized to layer $\mathrm{L}_1$ input. Distinct frequency regimes exhibit near-uniform motion, strong band gap attenuation, and weaker viscoelastic damping. B) Transmissibility $\mathrm{T}$ from optical vibrometry, along each axis and for each layer $\mathrm{L}_j$. Band gaps are shaded, with insets showing gap-averaged gains, fitted to an exponential decay as in Floquet-Bloch theory. C) Experimental setup for global transmission measurements using accelerometers. D) Density of the metamaterial and common silicones, either plain or mixed with micro glass bubbles. E) Transmissibility $\mathrm{T}^{\,\prime}$ from accelerometer data for the metamaterial either empty or filled, averaged over ten trials. F) Corresponding results for the reference silicones.