Geometry-driven filamentary structures : elastic gridshells, weaves, clasps, and knots

Abstract

In this thesis, we cover four research topics in the realm of the mechanics of slender structures involving strong geometric constraints: elastic gridshells, triaxial weaves, elastic clasps, and elastic knots. These studies involve a combination of geometric reasoning, high-fidelity numerical simulations, and precision model experiments using scale-invariance and advanced imaging techniques (e.g., 3D laser scanning, and X-ray computed tomography). First, we study the shape and the mechanical response of elastic gridshells, the three-dimensional structure of which results from the out-of-plane buckling of an initially flat and biaxial network of rods. A purely geometric continuum model, originally introduced by Chebyshev for woven fabric, is used to describe the underlying kinematics and form-finding. The results suggest that rod inextensibility, rather than elasticity, is the primary factor that determines the shape of elastic gridshells.

Second, we investigate triaxial weaving, a craft technique used to generate surfaces using tri-directional arrays of initially straight elastic ribbons. Traditional weavers intentionally introduce discrete topological defects, leading to unsmooth surfaces in the overall structure. As an alternative point of departure, we achieve smooth, threedimensional weaved structures by prescribing in-plane curvatures to the flat ribbons. We demonstrate that a continuous range of integrated Gaussian curvatures can be achieved, which is not feasible using straight ribbons. The potential of this novel design scheme is demonstrated with a few canonical target shapes.

Third, we investigate the mechanics of two elastic rods in a crossing contact, whose geometric counterpart is often referred to in the mathematics community as a 'clasp.' We compare our experimental and computational results to a well-established description for ideal clasps of geometrically rigid strings, finding that the latter acts as an underlying 'backbone' for the full elasticity solution. Our findings suggest that the tight contact between rods must be analyzed as a three-dimensional solid, not a one-dimensional rod. We also study a frictional elastic clasp with relative motion between the two rods. Finally, we present preliminary results on the full three-dimensional finite element method simulations of tight elastic knots, as a continuing discussion of tight contact between filaments. Our numerical results reveal significant deviations for the tight knots from existing one-dimensional models for loose overhand knots.

Our findings corroborate the three-dimensional nature of the tight contact that we demonstrated during the investigation of the elastic clasp.