A ruled narrow elastic strip model with corrected energy

TL;DR

This paper introduces a new one-dimensional elastic strip model based on nondevelopable ruled surfaces, which corrects and extends the Sadowsky model to include nonzero twist and improved energy convexity.

Contribution

The authors develop a novel ruled surface-based model with a regularized energy functional, allowing for nonzero twist and convexification of the Sadowsky energy without patching.

Findings

Energy exhibits Freddi et al.'s scalings

Convexifies the Sadowsky energy for certain parameters

Curvature and moment are continuous at inflection points when energy is convex

Abstract

We present a new one-dimensional model for elastic strips based on a nondevelopable ruled surface. An auxiliary field regularizes the Sadowsky narrow-strip model to allow nonzero twist with vanishing curvature. The energy exhibits the scalings derived by Freddi and co-workers, and for a certain choice of parameter, convexifies the Sadowsky energy without patching. We present the kinematics and energetics of the model, and employ a variational approach featuring a rotation tensor to derive equilibrium equations. We perform a regular perturbation expansion to study the model behavior close to inflection points. When the energy is convex, curvature and moment are continuous at inflection points, while the auxiliary function suffers a jump, leading to a discontinuity in the ruled embedding for any finite width.