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Selective and collective actuation in active solids

Abstract

Active solids consist of elastically coupled out-of-equilibrium units performing work1,2,3,4,5,6,7,8,9,10,11,12,13. They are central to autonomous processes, such as locomotion, self-oscillations and rectification, in biological systems14,15,16,17,18,19,20,21,22,23,24,25, designer materials26 and robotics27,28,29,30,31. Yet, the feedback mechanism between elastic and active forces as well as the possible emergence of collective behaviours in a mechanically stable elastic solid remains elusive. Here we introduce a minimal realization of an active elastic solid in which we characterize the emergence of selective and collective actuation resulting from the interplay between activity and elasticity. Polar active agents exert forces on the nodes of a two-dimensional elastic lattice. The resulting displacement field nonlinearly reorients the active agents. For a large-enough coupling, a collective oscillation of the lattice nodes around their equilibrium position emerges. Only a few elastic modes are actuated and crucially, they are not necessarily the lowest energy ones. By combining experiments with the numerical and theoretical analyses of an agent’s model, we unveil the bifurcation scenario and selection mechanism by which the collective actuation takes place. Our findings may provide a new mechanism for oscillatory dynamics in biological systems14,19,21,24 and the opportunity for bona fide autonomy in metamaterials32,33.

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Fig. 1: Active-solids design principle.
Fig. 2: Selective and collective actuation in two-dimensional elastic lattices, pinned at their edges.
Fig. 3: Large-N lattices.
Fig. 4: Mean-field, single-particle, one-dimensional-lattice phase diagrams.

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Acknowledgements

P.B. was supported by a PhD grant from ED564 ‘Physique en Ile de France’. D.S. was supported by a Chateaubriand fellowship. G.D. acknowledges support from Fondecyt grant no. 1210656 and ANID–Millenium Science Initiative Program—Code NCN17_092. We are grateful to M. Fruchart and V. Vitelli for fruitful discussions regarding non-reciprocity in elastic materials.

Author information

Authors and Affiliations

Authors

Contributions

O.D., C.C. and G.D. conceived the project. P.B. and D.S. performed the experiments. P.B., D.S. and O.D. analysed the experimental results. P.B., V.D., O.D., C.H.L. and G.D. worked out the theory. All the authors contributed to the writing of the manuscript.

Corresponding authors

Correspondence to P. Baconnier or O. Dauchot.

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The authors declare no competing interests.

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Nature Physics thanks Anton Souslov and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Fixed points stability thresholds for the experimental’s structures.

(a/c) Stability thresholds upper-bound \({\pi }_{c,u}^{\{j,k\}}\) computed from Eq. (S45) for every pair of modes. The darker the pixel, the greater the upper-bound found. Remarkably, the best bound is always achieved for the pair of modes concerned by the condensation. (b/d) Fraction of stable fixed points as a function of π. The fraction of stable fixed points (•) is computed by integrating the histogram of the stability thresholds found by Eq. (S36) on one million configurations of the polarity field, drawn randomly and independently. Inset: sketch of the two most excited modes, which realized the best bound in (a/c). (a/b) For the triangular lattice pinned at the edges (α = 1.29). \({\pi }_{c}^{\min }=0.676\), \({\pi }_{c}^{\max }=1.20\), \({\pi }_{c}^{{{{\rm{upp}}}}}=1.49\) (c/d) For the kagome lattice pinned at the edges (α = 1.02). \({\pi }_{c}^{\min }=0.375\), \({\pi }_{c}^{\max }=0.751\), \({\pi }_{c}^{{{{\rm{upp}}}}}=0.883\).

Extended Data Fig. 2 Effect of the noise on the collective actuation regime.

(a) Averaged (over time) mean angular frequency (over particles) in absolute value as a function of D for π = 2.0. The gray area represents the 1-σ fluctuations. (b) Averaged (over time) mean angular frequency (over particles) in absolute value as a function of π for increasing angular noise. From top to bottom, D = 0, 10−3, 10−2, 2.10−2, 5.10−2. Fluctuations are not shown for sake of clarity. (c) Mean (over particles) angular frequency as a function of time for π = 2.0 and D = 10−1 in the triangular lattice pinned at the edges.

Supplementary information

Supplementary Information

Supplementary Sections 1–10 and Figs. 1–12.

Supplementary Video 1

Alignment experiment: we impose a square motion to an active building block and look at the response of the polarity. At each side of the square, the polarity aligns towards the new velocity vector. Acquired at 40 fps and displayed in real time.

Supplementary Video 2

Experiment: translation in a triangular lattice (N = 37) with free-boundary condition. Acquired at 30 fps and displayed in real time.

Supplementary Video 3

Experiment: rotation dynamics in a triangular lattice (N = 37) with free-boundary condition. Acquired at 30 fps and displayed in real time.

Supplementary Video 4

Experiment: frozen disordered dynamics in the triangular lattice (N = 19 active units). Acquired at 40 fps and displayed in real time.

Supplementary Video 5

Experiment: collective actuation regime in the triangular lattice (N = 19 active units). Acquired at 40 fps and displayed in real time.

Supplementary Video 6

Experiment: heterogeneous regime in the triangular lattice (N = 19 active units). Acquired at 40 fps and displayed in real time.

Supplementary Video 7

Experiment: frozen disordered dynamics in the kagome lattice (N = 12 active units). Acquired at 40 fps and displayed in real time.

Supplementary Video 8

Experiment: collective actuation dynamics in the kagome lattice (N = 12 active units). Acquired at 40 fps and displayed in real time.

Supplementary Video 9

Experiment: frozen dynamics in the single-particle system. Acquired at 40 fps and displayed in real time.

Supplementary Video 10

Experiment: spontaneous oscillations in the single-particle system. Acquired at 40 fps and displayed in real time.

Supplementary Video 11

Numerical simulation: annealing in π in the triangular lattice (N = 19 active units), with the same tension as in the experiment shown in Supplementary Video 6. The elasto-active coupling π is decreased from the collective actuation regime until the system finds a fixed point. Individual polarities are represented by a black arrow, springs are colour coded by the stress state; an elongated spring turns red, whereas a compressed one turns blue.

Supplementary Video 12

Numerical simulation of a large triangular lattice (N = 1,141 active units), sized such that its lower-energy modes have squared eigenfrequencies equal to unity. The system is initialized with zero displacement in every node and random initial condition for the polarities orientations, and π = 2.0. Polarities are shown as arrows coloured by their orientations. Springs are represented in grey. The system quickly finds a collective actuation regime.

Supplementary Video 13

Numerical simulation of a large kagome lattice (N = 930 active units), sized such that its lower-energy modes have squared eigenfrequencies equal to unity. The system is initialized with zero displacement in every node and random initial condition for the polarities orientations, and π = 10.0. Polarities are shown as arrows coloured by their orientations. Springs are represented in grey. The system quickly finds a collective actuation regime.

Source data

Source Data Figs. 2–4 and Extended Data Figs. 1 and 2

Experimental and numerical source data.

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Baconnier, P., Shohat, D., López, C.H. et al. Selective and collective actuation in active solids. Nat. Phys. (2022). https://doi.org/10.1038/s41567-022-01704-x

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