Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Table of Contents - 2008 - Volume 138, Issue 04  


Convergence of equilibria of three-dimensional thin elastic beams

M. G. Mora a1 and S. Müller a2
a1 SISSA, Via Beirut 2–4, 34014 Trieste, Italy (mora@sissa.it)
a2 Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103 Leipzig, Germany (sm@mis.mpg.de)

Article author query
mora mg   [Google Scholar] 
muller s   [Google Scholar] 
 

Abstract

A convergence result is proved for the equilibrium configurations of a three-dimensional thin elastic beam, as the diameter $h$ of the cross-section tends to zero. More precisely, we show that stationary points of the nonlinear elastic functional $E^h$, whose energies (per unit cross-section) are bounded by $Ch^2$, converge to stationary points of the $\varGamma$-limit of $E^h/h^2$. This corresponds to a nonlinear one-dimensional model for inextensible rods, describing bending and torsion effects. The proof is based on the rigidity estimate for low-energy deformations by Friesecke, James and Müller and on a compensated compactness argument in a singular geometry. In addition, possible concentration effects of the strain are controlled by a careful truncation argument.

(Published Online July 28 2008)
(Received November 15 2006)
(Accepted June 6 2007)