Non-reciprocal buckling makes active filaments polyfunctional

TL;DR

The paper shows that breaking reciprocity in slender elastic beams via internal non-reciprocal torque coupling converts buckling into self-sustained shape cycles through a critical exceptional point (CEP). A continuum theory with an odd elasticity component and a minimal odd von Mises truss model reveals a Z2 symmetric Bogdanov–Takens structure organizing a SNIC transition between buckling and self-snapping, and the CEP governs the dynamics near the transition. Discrete simulations and a 1D robotic metamaterial filament powered by embedded motors demonstrate that environmental perturbations can trigger crawling, digging and walking modalities in a single self-contained filament. This work establishes non-Hermitian, non-reciprocal physics as a practical design principle for programmable active materials and soft robotic metamaterials with multiple functional modes.

Abstract

Active filaments are a workhorse for propulsion and actuation across biology, soft robotics and mechanical metamaterials. However, artificial active rods suffer from limited robustness and adaptivity because they rely on external control, or are tethered to a substrate. Here we bypass these constraints by demonstrating that non-reciprocal interactions lead to large-scale unidirectional dynamics in free-standing slender structures. By coupling the bending modes of a buckled beam anti-symmetrically, we transform the multistable dynamics of elastic snap-through into persistent cycles of shape change. In contrast to the critical point underpinning beam buckling, this transition to self-snapping is mediated by a critical exceptional point, at which bending modes simultaneously become unstable and degenerate. Upon environmental perturbation, our active filaments exploit self-snapping for a range of functionality including crawling, digging and walking. Our work advances critical exceptional physics as a guiding principle for programming instabilities into functional active materials.

Figures (15)

• Figure 1: Non-reciprocal Buckling. Adding non-reciprocity to the canonical Euler buckling scenario causes beams to lose stability and transition into self-snapping. (a) Within a building block, each linkage has a torque-angle relationship $\tau_i =k^\mathrm{o}(\delta \theta_{i+1} -\delta \theta_{i-1})$, and responds antisymmetrically when perturbed from the left vs. right. Buckling a chain of non-reciprocal linkages under clamped tangent boundary conditions, we find that as non-reciprocity $k^\mathrm{o}$ increases, the beam polarizes but retains stability. Increasing activity still further, this stable state disappears and the beam persistently self-oscillates. Traces indicate linkage positions over time. Scale bar $7.5\mathrm{cm}$. (b) Decomposing the horizontal position of each linkage into normalized Fourier modes $a_k$, we find that the lowest two modes dominate the snapping process, oscillating out of phase with one another. Modes $a_k$ for $k>2$ are shown in gray, and remain negligible during snapping. (c) Smooth waves of odd torques run down the filament during snapping. (d) A single snap proceeds via a pulsation traveling from the tail of the filament to its head. Schematic indicates the torque dipoles exerted by each motor, alongside the force on the central linkage.
• Figure 2: Breaking reciprocity in slender filaments creates one-way flexural waves. (a, b) Perturbing an open chain of non-reciprocal linkages, we observe one-way advection of curvature. Panel (a) shows key frames of a perturbation applied at one free end. Panel (b) shows a perturbation applied at the chain midpoint, confirming the asymmetry of wave propagation. Colors indicate angular deviations from the flat state. (c) Non-reciprocal forces on a sinusoidal filament show a $\pi/2$ phase lag with respect to the filament geometry itself, leading to an antisymmetric flexural wave dispersion relation. Here $\lambda(q)$ is the complex dispersion from Eq. \ref{['eq:RingDispersion']}, with $\mathrm{Im}(\lambda)$ representing wave propagation and $\mathrm{Re}(\lambda)$ giving growth or decay. Dispersion shown for $\beta=\eta=1$.
• Figure 3: The odd von Mises truss is a minimal model of non-reciprocal buckling.(a, b) We capture non-reciprocal buckling in an odd von Mises truss with clamped ends. We compress this truss a distance $\text{d}l$ from its rest length. As $k^\mathrm{o}$ increases, the truss polarizes, and then begins to self-snap. (a) The internal angles of the truss ($\theta_\mathrm{L},\theta_\mathrm{LC},\theta_\mathrm{RC},\theta_\mathrm{R}$) with experimental realization below. (b) The non-reciprocal torques at each vertex during a single snap-through. (c) We measure the self-snapping frequency $\omega$ as a function of $k^\mathrm{o}$ for a range of passive bending stiffness $B$: $B=8.5\text{ mNm}$ and $B=6.8\text{ mNm}$ (circles) are implemented via electronic feedback, with $B=20\text{ mNm}$ (squares) implemented via a silicone elastomer. Rescaling the activity by this bending rigidity, $k^\mathrm{o}/B$, and the snapping frequency $\omega$ by the characteristic dissipative timescale $\omega \Gamma /B$ collapses these data onto a single curve given by Eq. \ref{['eq:CEP']}. We find a square-root growth of frequencies $\omega\sim \sqrt{k^\mathrm{o}-k^\mathrm{o}_c}$ beyond a critical activity $k^\mathrm{o}_c$ that is characteristic of an exceptional transition. (d, e) Our theory Eq. \ref{['eq:TrussDynamics']} gives a complete phase diagram of the bifurcations in this minimal system. This shows a stable region (gray) and buckling phases: at lower compression a simple buckling (blue), then at higher compressions buckling with transient jackknifing (green), along with the snapping phase where the non-reciprocity creates a limit cycle in phase space (orange). Crucially these four regions encircle a critical exceptional point, CEP, red dot in panel (d). The purple and brown dots correspond to a cusp bifurcation and a global bifurcation where the narrow region of multi-stable buckling and snapping collapses to a global SNIC bifurcation.
• Figure 4: Polyfunctions of non-reciprocal filaments. The same filament exhibits different locomotion modalities in response to distinct environmental perturbations. (a, b) Buckling our filaments via a free-standing brace, unidirectional shape cycles enable crawling on a substrate. Key frames are shown in panel (a) (i-v), with projections onto the lowest shape modes in (b). First, the rear linkage descends, driving the chain forward (i, ii). The filament's anchor point then swaps to the front linkage (iii), as the rear retracts (iv) and the cycle repeats (v). Displacement measured in body lengths (BL). (c) Locomotion velocity and snapping frequency versus the non-reciprocity $k^{o}$. The filament switches gait from crawling to jumping as the snapping frequency reaches the timescale of vertical motion in the filament's centre of mass. (d) The active filament navigates an obstacle by spontaneously switching gaits. The trajectory of the centre of mass is shown here and colored by the snapping frequency. As the filament touches the obstacle and leaves the substrate, it switches to the jumping gait. Once the middle units touch the obstacle, the filament snaps slowly again and lifts its body to climb up the obstacle. Inset shows details of the transition where the filament jumps, climbs, then jumps over the obstacle. (e) A half-clamped filament spontaneously steps back and forth when driven into contact with a substrate. Inset shows a kymograph with the angle deviations at free vertices over time, which clearly shows unidirectional waves. (f) A standing active filament digs into a granular pile of steel beads. While compressing the active filament, it scoops steel beads and pulls them aside as it snaps. (g) Breaking left–right symmetry enables the standing filament to walk. We tilt the filament by an angle $\phi$ and allow the top end to slide freely. The filament walks persistently towards the tilting direction. (h) Walking velocity versus the tilt angle $\phi$.
• Figure S1: Phase portraits of the odd von Mises truss. Phase portraits of the dynamics of the odd von Mises truss Eq. \ref{['eq:EoMSI2']} in $\theta_\text{S}$ (horizontal) and $\theta_\text{A}$ (vertical) space, for different values of compression, $\text{d}l$, and non-reciprocity, $k^\text{o}$. Blue dots indicate fixed points. For a given $\text{d}l$, as $k^\mathrm{o}$ increases fixed points representing buckled states coalesce to form a limit cycle describing self-snapping.