Dynamical geometric modes in non-Euclidean plates

Continuum elastic shells with residual stresses can exhibit complex motions that require vanishing elastic energy to excite, also called geometric zero modes, 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. Plates with a curved, rather than flat, in-plane geometry, also called non-Euclidean plates, are one example in which residual stresses and geometry combine to produce a continuum of ground states. The plate can explore these states through a geometric zero mode when it has the shape of a minimal surface—a surface with zero mean curvature. We demonstrate that a non-Euclidean plate with the shape of a saddle 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.