Nonreciprocal dynamics in soft active structures: From swimming filaments to odd periodic systems

Reciprocity is a fundamental property of many physical systems, generally expressing a symmetry relation between two processes in which, for instance, input and output are interchanged. This remarkable and fruitful property underpins powerful theoretical and experimental techniques; at the same time, its tight constraints represent an obstacle to desired behaviours or functionalities. For instance, in low Reynolds number hydrodynamics, reciprocity forbids net thrust or flow generation through time-reversible deformations; microorganisms have hence adapted their motion so as to circumvent this constraint. Likewise, reciprocity in linear elasticity impedes the creation of wave-bearing media where input and output are not interchangeable, prompting recent research efforts aimed at finding means to break loose of this constraint. This thesis explores reciprocity breaking in soft active structures, focusing on two domains: biomimetic motility in soft robotics and wave propagation in periodic media. Active matter - whether biological or synthetic - negates reciprocity by converting stored or ambient energy into mechanical work. In living systems, this occurs in specialized motor proteins; in synthetic analogues, this is achieved through the use of smart materials, such as stimuli-responsive hydrogels. In these systems, internal activity introduces directed momentum input and disrupts the symmetries underlying classical reciprocity theorems. Using polyelectrolyte hydrogel filaments as a model system for one dimensional active structures, we show that flutter instability can be harnessed as a new mechanism for undulatory locomotion and periodic beating of active filaments requiring minimal actuation. Far from the detrimental consequences often associated to its advent, here flutter instability emerges from the interaction of the active filament with the fluid environment, thus allowing for the reduction of control complexity; its exploitation can hence be regarded as a form of embodied mechanical intelligence. From another perspective, the responsiveness of polyelectrolyte hydrogels to environmental cues, breaks reciprocity at the constitutive level, leading to direction-dependant wave bearing properties. We explore this phenomenology in a periodic metabeam composed of active rods, demonstrating nonreciprocal transmission in both static and dynamic regimes. The material's ability to harvest energy from external stimuli compensates viscous dissipation, enabling near-lossless unidirectional wave propagation in a fluid environment. This thesis represents a synthesis between mathematical modelling and physical experiments, in mutual and continuous dialogue. Theory and numerical simulations have informed the design and execution of the experiments, while laboratory observations have provided hints on how to refine the theoretical models and occasionally inspired new research directions.