The design space of plane elastic curves

Abstract

Elastic bending of initially flat slender elements allows the realization and economic fabrication of intriguing curved shapes. In this work, we derive an intuitive but rigorous geometric characterization of the design space of plane elastic rods with variable stiffness. It enables designers to determine which shapes are physically viable with active bending by visual inspection alone. Building on these insights, we propose a method for efficiently designing the geometry of a flat elastic rod that realizes a target equilibrium curve, which only requires solving a linear program. We implement this method in an interactive computational design tool that gives feedback about the feasibility of a design, and computes the geometry of the structural elements necessary to realize it within an instant. The tool also offers an iterative optimization routine that improves the fabricability of a model while modifying it as little as possible. In addition, we use our geometric characterization to derive an algorithm for analyzing and recovering the stability of elastic curves that would otherwise snap out of their unstable equilibrium shapes by buckling. We show the efficacy of our approach by designing and manufacturing several physical models that are assembled from flat elements.

Generalized deployable elastic geodesic grids

“…Introducing elements with variable stiffness furthermore enables a wide range of target equilibrium shapes. This space was recently rigorously characterized [Hafner and Bickel 2021] for elements that are only constrained at their boundaries, using a method to determine physically viable shapes by visual inspection. To find viable grid layouts, we face a similar task and build upon these insights.…”

“…We expect that all curves of type d) need additional support from crossing members. Curves of type c) are a limit case and are assessed using the criterion of [Hafner and Bickel 2021].…”

“…We consider members with 𝜎 < 2 as stable and members with 𝜎 > 2 as unstable, please refer to Figure 12 for a depiction. To classify members with 𝜎 = 2, we use a stability criterion for planar curves [Hafner and Bickel 2021] (neglecting the torsion of grid members). We subsequently classify each curve with a shape stability parameter 𝜂 = 0 or 𝜂 = 1.…”