Optimal design of odd elastic metamaterials

Non-reciprocal interactions in active solids yield elastic moduli forbidden in equilibrium. These odd moduli offer a bottom-up approach to designing autonomous materials that spontaneously crawl, roll or swim. However, current schemes for designing odd materials are typically overactuated, using excessively many active elements. This overactuation limits the feasibility of large-scale experimental realisations, and invites the challenge of rational design: how can microscopic non-reciprocity be optimally converted into macroscale odd response?



Here we show that odd moduli emerge in a broad range of lattices made of non-reciprocal springs. However, the strength of odd response strongly depends on the precise lattice geometry. Hyperstatic lattices are needlessly hard to actuate, leading to sub-optimal odd response. By contrast, we find that in overly floppy lattices, zero modes couple to microscopic non-reciprocity, destroying odd moduli entirely. By avoiding these pitfalls, we identify optimal design principles for building odd lattices.



Our results offer a blueprint for efficient engineering of odd lattices, opening the door to experimental realisations that truly probe the continuum limit of these robotic materials.