Dynamical geometric modes in non-Euclidean plates

TL;DR

This paper investigates the dynamical behavior of geometric zero modes in non-Euclidean elastic shells, revealing their soft, pendulum-like dynamics and resonance phenomena through theory and experiments.

Contribution

It introduces a combined analytical and experimental study of the elastodynamics of geometric zero modes in non-Euclidean plates, highlighting their pendulum-like behavior and resonance effects.

Findings

Zero modes exhibit pendulum-like dynamics.

Resonance phenomena include small oscillations and steady rotations.

Degeneracy is lifted by aging, but zero mode remains the softest.

Abstract

When subjected to specific prestresses, continuum elastic shells can exhibit geometric zero modes: complex motions that require vanishing elastic energy to excite, enabling them to be driven by weak and generic energy inputs. Despite recent interest in these modes, we understand very little about their dynamical properties. Non-Euclidean plates modeled on minimal surfaces are one example in which prestresses and geometry combine to produce a continuum of ground states that the plate can explore through a geometric zero mode. We demonstrate that a non-Euclidean plate with metric corresponding to Enneper's minimal surface exhibits the predicted continuous stability, but this degeneracy is ultimately lifted by aging. Despite developing a preferred configuration, the zero mode remains the softest mode. Using a combination of analytical theory and experiments, we show that the elastodynamics of this soft mode is captured by the dynamics of a damped pendulum. A periodic driving uncovers resonance phenomena in this pendulum mode, such as small oscillations and steady rotations, but mixes with an additional flapping mode at high frequencies.