On global and local minimizers of prestrained thin elastic rods

TL;DR

This paper derives a one-dimensional limit theory for prestrained elastic rods, analyzing stable configurations and comparing theoretical results with experiments, focusing on local and global minimizers.

Contribution

It introduces a new limit model for prestrained rods via $3$-convergence and explores stability of configurations, including simplified Kirchhoff models for specific cases.

Findings

Isolated local minimizers of the limit model can be approximated by those of the 3D model.

The limit energy simplifies to a Kirchhoff rod model for isotropic, two-layer prestrained rods.

Simulations align theoretical predictions with experimental observations.

Abstract

We study the stable configurations of a thin three-dimensional weakly prestrained rod subject to a terminal load as the thickness of the section vanishes. By $\Gamma$-convergence we derive a one-dimensional limit theory and show that isolated local minimizers of the limit model can be approached by local minimizers of the three-dimensional model. In the case of isotropic materials and for two-layers prestrained three-dimensional models the limit energy further simplifies to that of a Kirchhoff rod-model of an intrinsically curved beam. In this case we study the limit theory and investigate global and/or local stability of straight and helical configurations. Through some simple simulations we finally compare our results with real experiments.