A non-linear rod model for folded elastic strips

TL;DR

This paper develops a non-linear rod model for folded elastic strips, capturing large deformations and buckling behaviors, and identifies novel buckling patterns through theoretical analysis, experiments, and simulations.

Contribution

It introduces a new non-linear effective rod model with a geometrically constrained constitutive law for folded elastic strips, including the ridge angle as an internal degree of freedom.

Findings

Identified a buckling instability leading to out-of-plane configurations.

Discovered two novel buckling patterns: planar ridge modulation and localized bending.

Validated theoretical predictions with experiments and simulations.

Abstract

We consider the equilibrium shapes of a thin, annular strip cut out in an elastic sheet. When a central fold is formed by creasing beyond the elastic limit, the strip has been observed to buckle out-of-plane. Starting from the theory of elastic plates, we derive a Kirchhoff rod model for the folded strip. A non-linear effective constitutive law incorporating the underlying geometrical constraints is derived, in which the angle the ridge appears as an internal degree of freedom. By contrast with traditional thin- walled beam models, this constitutive law captures large, non-rigid deformations of the cross-sections, including finite variations of the dihedral angle at the ridge. Using this effective rod theory, we identify a buckling instability that produces the out-of-plane configurations of the folded strip, and show that the strip behaves as an elastic ring having one frozen mode of curvature. In addition, we point out two novel buckling patterns: one where the centerline remains planar and the ridge angle is modulated; another one where the bending deformation is localized. These patterns are observed experimentally, explained based on stability analyses, and reproduced in simulations of the post-buckled configurations.