Geometry and Mechanics of Leaves and the Role of Weakly-Irregular Isometric Immersions

Abstract

Thin elastic objects, including leaves, flowers, plastic sheets and sails, are ubiquitous in nature and their technological applications are growing with the introduction of hydrogel thin-films, flexible electronics and environmentally responsive gels. The intricate rippling and buckling patterns are postulated to be the result of minimizing an elastic energy. In this dissertation, we investigate the role of regularity in minimizing the elastic energy. Though there exist smooth isometric immersions of arbitrarily large subsets of H2 into R3, we show that the introduction of weakly-irregular singularities, of smoothness class C^{1,1}, significantly reduces the energy; we provide numerical evidence supporting an upper bound on the asymptotic scaling of the minimum energy over C^{1,1} isometries which is an exponentially large improvement as compared to the conjectured lower bound over C2 surfaces. This work provides insight into the quantitative nature of the Hilbert-Efimov theorem. The introduction of such singularities is energetically inexpensive, and so too is their relocation. Therefore, isometries are "floppy" or easily-deformable, motivating a shift in focus from finding the exact minimizers of the elastic energy in favor of understanding the statistical mechanics of the collection of zero-stretching immersions.