# One-dimensional von K\'arm\'an models for elastic ribbons

## TL;DR

This paper rigorously derives three one-dimensional models for elastic ribbons from von Kármán plate theory, capturing linear, nonlinear, and constrained behaviors as the plate width diminishes.

## Contribution

It introduces a rigorous variational derivation of three distinct one-dimensional ribbon models from two-dimensional von Kármán plate theory, including a new Sadowsky type model.

## Key findings

- Linear model describes elastic beams with twisting capability.
- Nonlinear model accounts for stretching, bending, and twisting.
- Constrained model leads to a novel Sadowsky type model.

## Abstract

By means of a variational approach we rigorously deduce three one-dimensional models for elastic ribbons from the theory of von K\'arm\'an plates, passing to the limit as the width of the plate goes to zero. The one-dimensional model found starting from the "linearized" von K\'arm\'an energy corresponds to that of a linearly elastic beam that can twist but can deform in just one plane; while the model found from the von K\'arm\'an energy is a non-linear model that comprises stretching, bendings, and twisting. The "constrained" von K\'arm\'an energy, instead, leads to a new Sadowsky type of model.