Nonreciprocal topological kink-wave propagation in mechanical metamaterials

Abstract

Nonlinear mechanical metamaterials can exhibit emergent transport phenomena that mimic topological protection without relying on linear band topology. Here, we realize a bifurcation-induced nonreciprocal lattice that supports robust propagation of elastic kink waves. Each unit is a prestrained, hinged-beam circulator that develops angular momentum bias during snap-through transitions between buckling states, producing an effective breaking of time reversal symmetry. Coupling such units into a hexagonal array yields a mechanically chiral network where localized soliton-like excitations propagate unidirectionally along interfaces and edges, immune to sharp bends. We demonstrate non-dispersive kink transport governed by a SineGordon type field whose effective bias encodes mechanical chirality. This framework bridges bifurcation dynamics and nonreciprocal transport, establishing a nonlinear route toward topological like mechanical functionality without magnetic or gyroscopic bias.

Figures (4)

• Figure 1: Mechanical circulator and reflector. (A) Straight flexible beam with pinned ends in secondary buckling (S-mode) snaps through to primary buckling (U-mode) under midpoint load (blue arrow). During snap-through, left hinge rotation $\alpha_l$ (magenta) exceeds right hinge rotation $\alpha_r$ (cyan), as shown versus normalized displacement ($u_1/a$) in the right panel. Beam: length $105~\mathrm{mm}$, width $10~\mathrm{mm}$, thickness $0.5~\mathrm{mm}$, hinge radius $20~\mathrm{mm}$, pre-compressed by $5~\mathrm{mm}$ to span $140~\mathrm{mm}$. (B) Mechanical circulator formed by three hinged beams in a triangular loop. Compression towards the centroid induces secondary buckling, producing a chiral three-port system. A load at Port 1 triggers $1\!\to\!2$ circulation (Port 2 active, Port 3 passive) via momentum biasing; similarly for $2\!\to\!3$ and $3\!\to\!1$. (C) Reflector created by adding a hard wall (purple) at Port 2, restricting motion inward. A load at Port 1 redirects displacement to Port 3 (mechanical reflection), while a load at Port 3 outputs directly to Port 1. (D) Port displacements ($u_1/a$): circulator—Port 2 (blue), Port 3 (black); reflector—Port 2 (green dashed), Port 3 (red dashed). Blue/yellow regions mark circulator pre/post-snap; unshaded/shaded mark reflector pre/post-snap. For $u_1>0$, responses coincide; for $u_1<0$, reflector snap-through is delayed. Middle: transmission before (magenta) and after (red) snap-through. Bottom: strain energy for circulator (blue) and reflector (red dashed). (E) Experimental circulator assembly into a mechanical topological metamaterial of two layers (I and II) connected via gemels; six co-rotational hinges (upper, middle, lower) prevent spatial interference.
• Figure 2: Topological Insulator. (A) Initial configuration of the mechanical topological domain wall Z path (left), deformed state (middle), and experimental snapshots at four time points (right group). Blue indicates clockwise circulators, green indicates counter-clockwise circulators, and yellow indicates deformed circulators. The white dashed line marks the interface between the two domains, and the grey arrow indicates the propagation direction. Stress distributions before and after deformation are shown in the left and middle images, respectively. Beams in the experiment were fabricated from thin spring steel. The parameters of the circulator are the same with Fig. 1. A total of 54 Circulators were used. (B) Mechanical topological domain wall circle path. Experimental parameters are identical to subfigure (A). (C) Mechanical topological edge state. A hard wall (purple block) was added at the medium boundary to reflect the displacement signal. Three boundary types were implemented: defect, armchair, and zigzag, to test the structure's defect immunity. A total of 41 circulators were used. All other parameters are identical to those stated above. Corresponding movies are presented in the supplementary videos S1 to S3.
• Figure 3: Topological soliton propagation. Experimental data for topological solitons propagating along the Z path (A), circle path (B), and the edge state (C). In panels A, B, and C, the horizontal axis represents time, the vertical axis represents the node coordinate index (excluding input and output nodes), with node numbering indicated in the insets. Blue indicates lower displacement, red indicates higher displacement. Propagation occurs at constant velocities: Z path $14.57~\mathrm{m/s}$, circle path $15.78~\mathrm{m/s}$, edge state $9.90~\mathrm{m/s}$. Grey lines indicate FEA simulation fits. The topological soliton waveforms for the Z path (D), circle path (E), and edge state (F) are also plotted. The horizontal axis is time, the vertical axis is soliton displacement. Dashed lines represent experimental results, solid lines represent FEA results. Soliton waveforms are shown for several nodes uniformly selected along the entire path. Stable soliton waveforms are maintained for the domain wall paths (Z path and circle path). For the edge state, the soliton waveforms show amplitude variation due to the pre-shifted hard wall, but no attenuation is observed. (G, H, I) To quantitatively verify soliton shape stability, Fourier transforms were performed on selected solitons to obtain their spectra. Comparison reveals highly consistent soliton spectra.
• Figure 4: Non-reciprocity of topological solitons. Forward excitation (top row) and reverse excitation (bottom row) were applied to the domain wall Z path (A), domain wall circle path (B), and edge state (C), respectively. The corresponding displacement bar graphs are plotted. Distinct propagation pathways are observed for forward and reverse directions, confirming the non-reciprocity inherent to the mechanical spin-like Hall effect.