Odd mechanics of active slender structures*

Living systems are chiral on multiple scales, from constituent biopolymers to large scale morphology, and their active mechanics is both driven by chiral components and serves to generate chiral morphologies. Increasingly such active chiral phenomena are being viewed through the lens of odd mechanics in both biological systems and in bio-inspired soft robotics. As many biological materials are localised to thin deformable interfaces such as membranes and filaments we consider the mechanics of active fluid membranes and active elastic rods, with a focus on odd contributions to the stress which break Maxwell-Betti reciprocity.

In the case of fluid membranes these generate geometric `odd elastic' forces in response to mean curvature gradients but directed perpendicularly. As a result, they induce tangential membrane flows that circulate around maxima and minima of membrane curvature. For shapes with anisotropy in the shape operator these flows can lead to linear instabilities. We describe examples for spheroids, membranes tubes and helicoids, discussing the relevance and predictions for left-right symmetry breaking in morphogenesis.

In the case of elastic rods we examine odd contributions to the bending moment which lead to unstable waves when inertia is non-negligible. We discuss the dispersion relations, coarsening dynamics and how these theories relate to experiments on odd robotic filaments.