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Odd elasticity

Abstract

A passive solid cannot do work on its surroundings through any quasistatic cycle of deformations. This property places strong constraints on the allowed elastic moduli. In this Article, we show that static elastic moduli altogether absent in passive elasticity can arise from active, non-conservative microscopic interactions. These active moduli enter the antisymmetric (or odd) part of the static elastic modulus tensor and quantify the amount of work extracted along quasistatic strain cycles. In two-dimensional isotropic media, two chiral odd-elastic moduli emerge in addition to the bulk and shear moduli. We discuss microscopic realizations that include networks of Hookean springs augmented with active transverse forces and non-reciprocal active hinges. Using coarse-grained microscopic models, numerical simulations and continuum equations, we uncover phenomena ranging from auxetic behaviour induced by odd moduli to elastic wave propagation in overdamped media enabled by self-sustained active strain cycles. Our work sheds light on the non-Hermitian mechanics of two- and three-dimensional active solids that conserve linear momentum but exhibit a non-reciprocal linear response.

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Fig. 1: Quasistatic energy cycles with non-conservative active bonds.
Fig. 2: Odd-elastic engine cycle.
Fig. 3: Static response in an odd-elastic solid.
Fig. 4: Odd-elastic waves.
Fig. 5: Exceptional points and non-Hermitian elastodynamics.

Data availability

The data represented in Fig. 3c are available as Source Data Fig. 3. All other data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.

Code availability

The code used to perform and analyse the numerics in this work is available from the corresponding author upon reasonable request.

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Acknowledgements

V.V. was supported by the Complex Dynamics and Systems Program of the Army Research Office under grant no. W911NF-19-1-0268. V.V., A.S. and W.T.M.I. acknowledge primary support through the Chicago MRSEC, funded by the NSF through grant no. DMR-1420709. A.S. acknowledges the support of the Engineering and Physical Sciences Research Council (EPSRC) through New Investigator Award no. EP/T000961/1. C.S. was supported by the National Science Foundation Graduate Research Fellowship under grant no. 1746045. W.T.M.I. acknowledges support from NSF EFRI NewLAW grant no. 1741685 and NSF DMR 1905974. D.B. was supported by FOM and NWO. P.S. was supported by the Deutsche Forschungsgemeinschaft via the Leibniz Program. We thank R. Lakes, F. Jülicher and G. Salbreux for their critical readings of the manuscript.

Author information

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Authors

Contributions

V.V. initiated the research. C.S., A.S., W.T.M.I. and V.V. prepared the manuscript. All authors conducted the research, revised the manuscript and contributed to discussions.

Corresponding author

Correspondence to Vincenzo Vitelli.

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Competing interests

The authors declare no competing interests.

Additional information

Peer review statement Nature Physics thanks Roderic Lakes and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Active hinge model.

a. A honeycomb plaquette with active hinges at each vertex. Each hinge exerts an angular tension Ti based on the angular strain δθi−1 of its counterclockwise neighbor. b. A quasistatic, strain-controlled cycle in which the plaquette does work on its surroundings.

Supplementary information

Supplementary Information

Supplementary Figs. 1–10 and Discussion.

Supplementary Video 1

A microscopic illustration of a non-conservative active bond.

Supplementary Video 2

Mechanics of an overdamped odd-elastic wave.

Supplementary Video 3

Simulation of odd-elastic waves.

Supplementary Video 4

Exceptional points in odd-elastic solids.

Source data

Source Data Fig. 3

Numerical data for plots in panel c.

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Scheibner, C., Souslov, A., Banerjee, D. et al. Odd elasticity. Nat. Phys. 16, 475–480 (2020). https://doi.org/10.1038/s41567-020-0795-y

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