A Dynamical Zero Mode in Non-Euclidean Plates

Frustrated thin elastic plates that try to achieve flat configurations while having a curved intrinsic geometric, also called non-euclidean plates, are known to exhibit rich physical phenomena, such as an anomalous soft elastic response, fractional solitons, and programmable shapes. Remarkably, the elastic energy of these plates presents a continuous symmetry when their intrinsic geometry is that of a minimal surface. By leveraging the zero (Goldstone) mode associated with this symmetry, we construct an effective 2D elasticity theory to explore wave propagation and the dynamics of these plates under external stimuli. We find that this zero-mode follows the dynamics of a pendulum, opening the possibility of exploring resonance phenomena. In the case of the Enneper minimal surfaces, the mode is localized to the boundary of the plate, akin to an edge mode. This understanding could help design responsive materials based on this zero mode.