Computation, Discrete Geometry, and Soft Mechanics in Non-Euclidean Elasticity

This dissertation explores the ways in which the geometry of thin objects influences their mechanics, i.e., the way they respond to external forces/stimuli and the ways that they move. This is relevant to biological phenomena, e.g., in the deployment of leaves, blooming of flowers, swimming of sea slugs, etc. It is also relevant to modern technological applications of soft materials including flexible and wearable electronics. We argue that the soft mechanics and dynamics of these non-Euclidean elastic sheets are governed by interacting non-smooth geometric defects in the material. Novel ideas stemming from characterizing and modeling these defects using Discrete Differential Geometry (DDG) are presented in order to uncover fundamental insights into the elastic behavior and properties of thin hyperbolic bodies, notably the inherent floppiness of these systems. In particular, we investigate, both analytically and numerically, the energetic impacts from non-smooth defects, the role of weak external forces, and associated scaling laws. The mathematics of the DDG formulation and implementation for modeling hyperbolic sheets is also derived and described. Finally, we connect our theory with experiments by presenting Bayesian techniques for analyzing noisy profilometric data for real-world sheets as well as predictions of buckling transitions of a hyperbolic gel compressed between two plates. New theories based on the mechanics of non-smooth defects may (i) explain biological phenomena, from the morphogenesis of leaves, flowers, etc. to the biomechanics of sea slugs, as well as (ii) introduce new paradigms for materials design and actuation in a variety of new technologies, e.g., soft robotics