The formation of rippled edges in leaves

Abstract

Differential in-plane growth of thin elastic sheets induces out-of-plane buckling. More growth near an edge produces rippled edges such as in plant leaves, underwater algae blades, and wet paper. Here, we combine a theory, finite element simulations, and experiments to analyze the relationship between the differential growth and the shape of rippled edges. First, we develop a theoretical model that predicts the formation of wave buckling in thin plates using the strain energy method. Then, finite element simulations are conducted to validate the theoretical model for various in-plane loading conditions. Experiments examine how a thin elastomeric sheet undergoes buckling instability when non-uniform stress is induced by swelling due to the diffusion of silicone oil. We measure time-dependent amplitude and wavelength of rippled edge and find good agreement with theoretical results. Finally, we conduct parametric exploration to relate the normalized amplitude and wavelength with the aspect ratio of sheets and growth in a wide range of scales. Our results can be used in biomimetic design where thin sheets of active materials are actuated with environmental stimuli such as heat, light, and concentration.