Transition Waves in Mechanical Metamaterials with Neighbor-Programmable Energy Landscapes

Abstract

Transition waves in mechanical metamaterials manifest themselves as propagating interfaces between different stable states in lattices composed of arrays of coupled, intrinsically bistable elements. Here, we show experimentally and numerically that arrays of elastic unit cells that are individually monostable, yet whose energy landscapes can be programmed through interactions with neighboring units, provide a rich and largely unexplored platform for transition wave propagation. We implement this concept by designing a unit cell comprising a von Mises truss supported by two vertical elastic beams. In one-dimensional arrays of such units, we demonstrate that each cell's energy landscape can change from monostable to bistable depending on the state of its neighbors. This neighbor-programmable energy landscape enables the controlled initiation and propagation of transition waves, giving rise to highly discrete, directionally unbiased, domino-like wave propagation. Experiments and numerical simulations show that the existence and speed of the waves are governed by geometric design and mass distribution. Our results establish neighboring effects as a distinct mechanism for transition wave propagation, expanding the design space of mechanical metamaterials beyond architectures that rely on intrinsically multistable building blocks.

Figures (21)

• Figure 1: Transition waves in mechanical metamaterials with neighbor-programmable energy landscapes.a) Schematic of the unit cell. b) The von Mises truss exhibits two distinct stable equilibrium states: the natural, stress-free up state and the inverted down state. c) Experimental snapshots of the mechanical characterization tests. A displacement is applied to the pull tab of the central unit while (i) all neighboring units are in the up state (red), (ii) the left neighbors are in the down state and the right neighbors in the up state (green), and (iii) all neighbors are in the down state (blue). d) Experimentally measured force–displacement curves. e) Corresponding energy landscapes. f) Sequential snapshots showing a transition wave propagating along the 32-unit metamaterial at four representative time points. g) Experimental (solid lines) and numerical (dashed lines) results for the temporal evolution of the vertical displacement $v^i_{\text{top}}$ of selected units $i$. h) Spatiotemporal displacement diagram. Supplementary Video 1 provides the corresponding experimental and numerical footage.
• Figure 2: Effect of geometry on wave propagation. Spatiotemporal displacement diagrams for the four geometries investigated, overlaid with the corresponding numerical predictions (red lines). Supplementary Video 2 presents the associated experimental recordings.
• Figure 3: Systematic exploration of the design space.a) Schematic of the mass–spring model. b) Numerically predicted evolution of the energy barrier $\Delta U$ as a function of $k_\theta$ and $k_\text{beam}$. c) Numerically predicted evolution of the wave speed $c_{\text{wave}}$ as a function of $k_\theta$ and $k_\text{beam}$. Red and blue markers indicate the four experimental structures considered in Fig. \ref{['fig:fig3']}, with labels (a–d) corresponding to the panels shown there.
• Figure 4: On-demand tuning of the wave speed. All results correspond to structures with $h = 0.7$ mm and $w = 1.4$ mm. a) Numerically predicted and experimentally measured evolution of the wave speed $c_{\text{wave}}$ as a function of the vertical pull-tab mass $m_{\text{top}}$. The reference experimental data point is highlighted with a circle. b) Spatiotemporal displacement diagram for a structure with two distinct masses: $\forall j \in [1,17],\, m^j_{\text{top}} = 0.52$ g, and $\forall j \in [18,32],\, m^j_{\text{top}} = 1.41$ g (see Supplementary Video 3).
• Figure S1: Geometry of the considered structures.