Topological Solitons Make Metamaterials Crawl

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Topological Solitons Make Metamaterials Crawl

Bolei Deng, Mohamed Zanaty, Antonio E. Forte, and Katia Bertoldi

Phys. Rev. Applied 17, 014004 – Published 4 January 2022

Abstract

In recent years, the ability to propagate topological solitons in mechanical metamaterials has unlocked unpaved paths towards potential applications in wave propagation, mechanical logic, and shape morphing. Here, we demonstrate how a multistable metamaterial can harness topological solitons with coupled rotational and translational components and become, itself, a crawling robot. We start by characterizing the topological solitons via experimental measurements and analysis. We then use their rotational component to produce a favorable gradient of friction between the metamaterial and the underlying substrate. This, in turn, creates locomotion. Previously proposed crawling robots usually require complex control of multiple actuators. In contrast, our robot can be powered by a single actuator, and all features needed for locomotion are embedded in the mechanics and activated by the topological solitons.

Received 7 June 2021
Revised 16 September 2021
Accepted 19 November 2021

DOI:https://doi.org/10.1103/PhysRevApplied.17.014004

Physics Subject Headings (PhySH)

Acoustic metamaterials Acoustic wave phenomena Classical mechanics Geometric & topological phases Locomotion Nonlinear acoustics Solitons Topological defects

General Physics

Authors & Affiliations

Bolei Deng ^1,†, Mohamed Zanaty^1,†, Antonio E. Forte^1,2,3, and Katia Bertoldi ^1,*

¹Harvard John A. Paulson School of Engineering and Applied Sciences Harvard University, Cambridge, Massachusetts 02138, USA
²Department of Electronics, Information and Bioengineering, Politecnico di Milano, Milan 20133, Italy
³Department of Engineering, King’s College London, London WC2R 2LS, United Kingdom

^*bertoldi@seas.harvard.edu
^†B. Deng and M. Zanaty contributed equally to this work.

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Vol. 17, Iss. 1 — January 2022

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Figure 1
Multistable mechanical metamaterial. (a) Bistable unit. (b) Elastic energy of the bistable unit, $E^{intra}$ , as predicted by experiments (solid line) and the discrete model (dashed line). (c) Snapshots of the bistable unit at equilibrium ( $θ_{j} = \pm θ_{st}$ ) and unstable ( $θ_{j} = 0$ ) states. (d) Schematics of the model. (e) Strain along the $x$ direction, $ϵ_{j}^{x}$ , as a function of rotation of the crosses, $θ_{j}$ , for the bistable unit. Notably, rotation extracted from experiments is the average of that of the top crosses. (f) 1D chain comprising four bistable units connected by elastic hinges of length $l_{h}^{inter}$ .
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Figure 2
Topological solitons in 1D chains. (a) Experimental snapshots of a metamaterial comprising $1 \times 5$ bistable units connected with hinges of length $l_{h}^{inter} = 6$ mm at $t = 0$ , 2.5, and 3.0 s. Actuator is mounted on the leftmost unit ( $j = 1$ ), the stroke-time relationship of which is shown in the inset. (b) Experimentally measured (solid lines) and numerically predicted (dashed lines) evolution of rotation (top) and $x$ -direction strain (bottom) of all bistable units as functions of time. (c) Rotation (top) and $x$ -direction displacement (bottom) of every unit in the chain as a function of position at three different times, as measured in experiments (triangles) and predicted by the discrete (circles) and continuum (solid lines). Rotation of every unit for chains with (d) $l_{h}^{inter} = 4$ mm and $N_{x} = 5$ , (e) $l_{h}^{inter} = 8$ mm and $N_{x} = 5$ , and (f) $l_{h}^{inter} = 8$ mm and $N_{x} = 4$ . Notably, reported results for the continuum model are obtained by adjusting $ξ_{0}$ in Eq. (10) to best match the corresponding results for the discrete model. Maps showing combinations of $N_{x}$ and $l_{h}^{inter}$ , resulting in pulses that reach the opposite end (yellow region for numerical results and green circular markers for experiments) or fail to switch all units (blue region for numerical results and red cross markers for experiments) in chains comprising units with (g) $θ_{st} = 30^{\circ}$ , (h) $θ_{st} = 25^{\circ}$ , and (i) $θ_{st} = 35^{\circ}$ . White line corresponds to the prediction of Eq. (13).
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Figure 3
Locomotion via topological solitons. (a) Bistable unit with two wheels mounted under each cross separated by distance $d_{wh}$ . (b) Magnitude of the friction force along the $x$ direction, $F_{j}^{fric, x}$ , as a function of the crosses’ rotation, $θ_{j}$ . (c) Experimental snapshots of a wheeled chain with $N_{x} = 3$ and $l_{h}^{inter} = 6$ mm at $t = 0$ , 1.8, 3.5, 5.0, and 15.0 s. (d) Rotation, $θ_{j}$ , and displacement, $u_{j}$ , of all units at $t = 0$ , 1.8, 3.5, 5.0, and 15.0 s, as measured in experiments (triangles) and predicted by the discrete model (circles). (e) Effect of $l_{h}^{inter}$ on step length $d_{step}$ , as measured in experiments (squares) and predicted by the discrete model (solid line). (f) Effect of $N_{x}$ and $l_{h}^{inter}$ on $d_{step}$ . Circles correspond to the chains considered in Figs. 3 and 4. White solid lines indicate analytically predicted combinations of parameters resulting in $N_{x} = W$ and $N_{x} = N_{c}$ . (g) Effect of $θ_{st}$ (which is determined by the rubber band’s initial length, $l_{b}$ ) on $d_{step}$ in a chain with $N_{x} = 3$ and $l_{h}^{inter} = 6$ mm.
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Figure 4
Wheeled chain with two actuators. (a) Experimental snapshots of a wheeled chain with $N_{x} = 6$ , $l_{h}^{inter} = 6$ mm and two actuators connected to the leftmost and rightmost units at $t = 0$ , 5, and 10 s. (b) Stroke of two actuators as a function of $t$ ; $t_{a}$ corresponds to the actuation time and $Δ t$ to the difference between the times at which the two actuators are activated. (c) Evolution of step length, $d_{step}$ , as a function of normalized time shift, $τ$ , as measured in experiments (squares) and predicted by the discrete model (solid line). (d) When hinges connecting units $j = 3$ and $j = 4$ fail, the chain splits into two independent and functional systems.
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Figure 5
Locomotion of 2D structures. (a) Experimental (top) and numerical snapshots showing a $2 \times 2$ structure with $l_{h}^{inter}$ = 5, 7, and 15 mm and one actuator connected to the top-left unit after 5 cycles of actuation. Dashed squares indicate their initial positions. (b) Effect of $l_{h}^{inter}$ on $d_{step}$ and global rotation, $β_{step}$ , for a $2 \times 2$ metamaterial with an actuator connected to the top-left unit. (c) Experimental (top) and numerical snapshots (bottom) showing a $2 \times 2$ structure with $l_{h}^{inter} = 11$ mm and two actuators connected to the left units after 5 cycles of actuation for different normalized time shifts of $τ = 0$ , 0.3, and $- 0.3$ . Dashed squares indicate their initial positions. (d) Effect of $τ$ on $d_{step}$ and $β_{step}$ for a $2 \times 2$ metamaterial with $l_{h}^{inter} = 11$ mm and actuators connected to both units on the left. (e) By varying $τ$ , we can guide the structure along complex paths. Dashed squares indicate their intermediate configurations, and yellow arrows represent their trajectories reached upon actuation. Notably, we present only the final configurations of the structure and use dashed lines to indicate its intermediate positions. Since the metamaterial returns to its initial phase after each cycle, the final configuration looks the same as the initial one, except for a rigid-body shift.
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