Convergence of equilibria of thin elastic rods under physical growth conditions for the energy density

TL;DR

This paper investigates how the equilibrium shapes of very thin elastic rods behave as their diameter shrinks, establishing convergence to various classical rod models under specific physical and loading conditions.

Contribution

It provides rigorous convergence results for the equilibrium configurations of thin elastic rods to classical models, under physically realistic energy density growth conditions.

Findings

Convergence to constrained linear theory

Convergence to von Kármán-like rod theory

Convergence to linear elastic theory

Abstract

The subject of this paper is the study of the asymptotic behaviour of the equilibrium configurations of a nonlinearly elastic thin rod, as the diameter of the cross-section tends to zero. Convergence results are established assuming physical growth conditions for the elastic energy density and suitable scalings of the applied loads, that correspond at the limit to different rod models: the constrained linear theory, the analogous of von K\'arm\'an plate theory for rods, and the linear theory.