Mechanical hysterons with tunable interactions of general sign

TL;DR

The work presents a physical platform of rigid bars and springs that realizes the abstract hysteron model with tunable, potentially non-reciprocal interactions, enabling designed non-equilibrium behaviors in mechanical metamaterials. A torque-balance (kinematic) framework maps geometry to switching thresholds $\gamma_i^\pm$ and interaction strengths $J_{ij}^\pm$, capturing both ferromagnetic-like and antiferromagnetic-like couplings, including non-reciprocity $J_{ij}^\pm \neq J_{ji}^\pm$. The authors demonstrate programmable transition graphs for two hysterons, realize latching via frustrated non-reciprocal interactions, and implement computations such as counting driving cycles and domain-wall ratcheting in larger networks, illustrating memory and computation in designed materials. This approach provides a general, experimentally accessible route to materials whose response encodes driving history, with potential applications in programmable robotics and smart sensing.

Abstract

Hysterons are elementary units of hysteresis that underlie many complex behaviors of non-equilibrium matter. Because models of interacting hysterons can describe disordered matter, this suggests that artificial systems could respond to mechanical inputs in precise and targeted ways. Specifying the properties of hysterons and their interactions could thus be a general method for realizing arbitrary non-equilibrium behaviors. Elastic structures including slender beams, creased sheets, and shells are clear candidates for artificial hysterons, but complete control of their interactions has seemed impractical or impossible. Here we report a mechanical hysteron composed of rigid bars and linear springs, which has controllable properties and tunable interactions of general sign that can be reciprocal or non-reciprocal. We derive a mapping from the system parameters to the hysteron properties, and we show how collective behaviors of multiple hysterons can be targeted by adjusting geometric parameters on the fly. By transforming an abstract hysteron model into a physical design platform, our work demonstrates a route toward designed materials that can sense, compute, and respond to their mechanical environment.

Figures (8)

• Figure 1: Coupled mechanical hysterons with tunable interactions. A. Each hysteron is a rigid bar that may freely rotate between two hard boundaries, under the influence of a driving spring that attaches to a bar applying a horizontal quasistatic global drive, $\gamma$. Hysterons may be coupled together with additional springs. B. Typical force-versus-displacement data for springs used in the experiment. Solid lines are linear fits used to extract the stiffness $k$ and rest length, $x_0$. C. Basic behavior of a hysteron. The thresholds, $\gamma^-$, $\gamma^+$, depend on the geometric properties of the system as well as the state of the other hysterons when interactions are present. D. Experimental realizations with cooperative (left) and frustrated interactions (right).
• Figure 2: Switching thresholds for an interacting hysteron pair. A. Using the configuration shown in the left panel of Fig. \ref{['fig:1']}D, we measure the eight switching thresholds as the mounting position of the coupling springs, $\ell$, is varied. Experimental uncertainty is $\pm 0.1$ cm, which is approximately the size of the symbols. The experiments are in good agreement with the theory. Geometric parameters: $y = 9.4$ cm, $L = 6.3$ cm, $w = 15.2$ cm, $x_1 = 0$ cm, $x_2 = -1.9$ cm, $[\theta_1^-,\theta_1^+] = [-36.6^\circ, 36.7^\circ]$, and $[\theta_2^-,\theta_2^+] = [-9.75^\circ,11.2^\circ]$. Driving springs: $k_1 = k_2 = 0.19$ N/cm, rest lengths $3.9$ and $4.2$ cm. Coupling springs: $k_{12} = 0.07$ N/cm, rest lengths $2.4$ and $2.6$ cm. B. Transition graphs showing behavior for $\ell < 4.18$ cm and $4.18 < \ell < 4.90$ cm. C. Splitting between pairs of switching thresholds, $\gamma_i^\pm(+) - \gamma_i^\pm(-)$, which measures the strength of interactions between hysterons. Triangles: Experiments. Solid lines: Kinematic model. Dashed lines: Eq. \ref{['eq:Jij']} for rotors confined to small angles.
• Figure 3: Using a frustrated non-reciprocal interaction to build a hysteron latch. A. Coupled mechanical hysterons that are stable in the $+-$ and $--$ states at $\gamma=0$. Driving up to $\gamma=10$ cm and back to $0$ sets the system in $+-$. Driving up to $\gamma=32$ cm and back to $0$ resets the system to $--$. Geometric parameters: $y = 9.4$ cm, $L = 6.3$ cm, $\ell=4.7$ cm, $w = 15.2$ cm, $x_1 = -8.1$ cm, $x_2 = -16.4$ cm, $\theta_1^\pm = \pm18.3^\circ$, and $\theta_2^\pm = \pm38.8^\circ$. All springs have $k = 0.19$ N/cm and $x_0 = 4.1$ cm. B. Switching thresholds in the experiment and the model. Arrows identify the (negative) values of $2J_{12} = \gamma_1^-(-) - \gamma_1^-(+)$ and $2J_{21} = \gamma_2^-(-) - \gamma_2^-(+)$, highlighting the frustrated non-reciprocal interaction at the core of the behavior. Experimental uncertainty is $\pm 0.1$ cm (not plotted).
• Figure 4: Counting driving cycles with a chain of mechanical hysterons. A. One-dimensional chain of hysterons with uniform antiferromagnetic nearest-neighbor interactions. All hysterons have the same hysteresis, $\gamma^+_i - \gamma^-_i$, but the thresholds are shifted positively/negatively for odd/even $i$ (dark/light gray in schematic). B. Time series showing a domain wall ratcheting down the chain, irreversibly flipping one hysteron each half cycle. The system is initialized in the state $+-$$+-$$+-$ and a cycle of driving follows $\gamma: \gamma_0 \rightarrow -\gamma_0 \rightarrow \gamma_0$. The growing $-+$$-+$$...$ phase is highlighted in blue. For ease of construction, the radial bearings are replaced by optical posts in post holders lubricated with silicone oil. Coupling springs: $k=0.17$ N/cm, $x_0 = 4.0$ cm. Driving springs: $k=0.19$ N/cm, $x_0 = 4.1$ cm.
• Figure 5: Counting modulo 2 with four mechanical hysterons. A. Thresholds and interactions that yield a period-2 limit cycle for unit-amplitude cyclic drive. B. Corresponding pathway, where we have colored the two states at the minimum of the driving. Lighter arrows show all other transitions available to the system under arbitrary driving and are not used in the period-2 cycle. States without arrows to them are unstable at all $\gamma$. C. Realizing this pathway with mechanical hysterons. The rotors from left to right correspond to the hysterons from top to bottom in A. The springs connecting rotors 1 and 3 have $k=0.07$ N/cm, and their original $x_0 = 2.5$ cm are extended by steel wire. All other springs have $k=0.19$ N/cm and $x_0 = 4.1$ cm. D. States and transitions in the experiment, which achieves the targeted response. Horizontal segments denote avalanches.