Technical Brief: Finite Element Modeling of Tight Elastic Knots

We present a methodology to simulate the mechanics of knots in elastic rods using geometrically nonlinear, full three-dimensional (3D) finite element analysis. We focus on the mechanical behavior of knots in tight configurations, for which the full 3D deformation must be taken into account. To set up the topology of our knotted structures, we apply a sequence of prescribed displacement steps to the centerline of an initially straight rod that is meshed with 3D solid elements. Self-contact is enforced with a normal penalty force combined with Coulomb friction. As test cases, we investigate both overhand and figure-of-eight knots. Our simulations are validated with precision model experiments, combining rod fabrication and X-ray tomography. Even if the focus is given to the methods, our results reveal that 3D deformation of tight elastic knots is central to their mechanical response. These findings contrast to a previous analysis of loose knots, for which 1D centerline-based rod theories sufficed for a predictive understanding. Our method serves as a robust framework to access complex mechanical behavior of tightly knotted structures that are not readily available through experiments nor existing reduced-order theories

The shapes of physical trefoil knots

We perform a compare-and-contrast investigation between the equilibrium shapes of physical and ideal trefoil knots, both in closed and open configurations. Ideal knots are purely geometric abstractions for the tightest configuration tied in a perfectly flexible, self-avoiding tube with an inextensible centerline and undeformable cross-sections. Here, we construct physical realizations of tight trefoil knots tied in an elastomeric rod, and use X-ray tomography and 3D finite element simulation for detailed characterization. Specifically, we evaluate the role of elasticity in dictating the physical knot's overall shape, self-contact regions, curvature profile, and cross-section deformation. We compare the shape of our elastic knots to prior computations of the corresponding ideal configurations. Our results on tight physical knots exhibit many similarities to their purely geometric counterparts, but also some striking dissimilarities that we examine in detail. These observations raise the hypothesis that regions of localized elastic deformation, not captured by the geometric models, could act as precursors for the weak spots that compromise the strength of knotted filaments

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

Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mechanical Engineering, February, 2021Cataloged from the official PDF of thesis.Includes bibliographical references (pages 217-232).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.by Changyeob Baek.Ph. D.Ph.D. Massachusetts Institute of Technology, Department of Mechanical Engineerin

Rigidity of hemispherical elastic gridshells under point load indentation

We present results from physical and numerical experiments on the rigidity of hemispherical gridshells under point load indentation. By systematically exploring the relevant parameters of the system, we provide a scaling law for the rigidity of elastic gridshells in terms of the dimension of the structure and the number of rods it contains, as well as the geometric and material properties of the individual rods. Our approach combines a set of precision desktop-scale experiments and discrete elastic rod simulations, which are found to be in excellent quantitative agreement. Our proposed empirical relation for the rigidity also points to the underlying nonlocal nature of the mechanical response of gridshells, in contrast to the local response of isotropic continuum shells. We further assess this nonlocality by quantifying the resulting radial displacement field as well as inspecting the effect of the location of the indentation point on the rigidity. (C) 2018 Elsevier Ltd. All rights reserved

Form finding in elastic gridshells

Elastic gridshells comprise an initially planar network of elastic rods that are actuated into a shell-like structure by loading their extremities. The resulting actuated form derives from the elastic buckling of the rods subjected to inextensibility. We study elastic gridshells with a focus on the rational design of the final shapes. Our precision desktop experiments exhibit complex geometries, even from seemingly simple initial configurations and actuation processes. The numerical simulations capture this nonintuitive behavior with excellent quantitative agreement, allowing for an exploration of parameter space that reveals multistable states. We then turn to the theory of smooth Chebyshev nets to address the inverse design of hemispherical elastic gridshells. The results suggest that rod inextensibility, not elastic response, dictates the zeroth-order shape of an actuated elastic gridshell. As it turns out, this is the shape of a common household strainer. Therefore, the geometry of Chebyshev nets can be further used to understand elastic gridshells. In particular, we introduce a way to quantify the intrinsic shape of the empty, but enclosed regions, which we then use to rationalize the nonlocal deformation of elastic gridshells to point loading. This justifies the observed difficulty in form finding. Nevertheless, we close with an exploration of concatenating multiple elastic gridshell building blocks.National Science Foundation (U.S.) (Grant CMMI-1351449