Geometric defects, weak forces, and self-similar buckling in non-Euclidean elastic sheets

Non-Euclidean elastic sheets (like lettuce, flowers, and sea slugs) exhibit extreme properties including an inherent floppiness, which we argue is governed by and may, in turn, be quantified by non-smooth geometric defects. The presence and interaction of these localized defects in hyperbolic sheets may be modelled and explored with rough solutions via discrete differential geometry (DDG) based on taking a singular limit that assumes a no-stretching constraint. Non-smooth geometric defects are identified by a skeleton of intersecting asymptotic curves which, along with geometric data along the curves, enables constructing complex morphologies with intricate wrinkling shapes. By computing various energies on discretized elements, numerical simulations using this novel DDG framework reveal the significant impact of non-smooth geometric defects on elastic energy as well as the non-negligible role of weak forces, i.e., effects other than stretching or bending, and associated scaling laws. Ultimately, these modeling techniques have the potential to explain real-world sheets and enable the control/design of thin soft structures.