Calculus of Variations and Geometric Measure Theory

M. G. Mora - S. Müller

Convergence of equilibria of three-dimensional thin elastic beams

created by mora on 14 Nov 2006
modified on 19 Sep 2008


Published Paper

Inserted: 14 nov 2006
Last Updated: 19 sep 2008

Journal: Proc. Roy. Soc. Edinburgh Sect. A
Volume: 138
Pages: 873-896
Year: 2008


A convergence result is proved for the equilibrium configurations of a three-dimensional thin elastic beam, as the diameter $h$ of the cross-section goes 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 $C h^2$, converge to stationary points of the Gamma-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.

Keywords: dimension reduction, nonlinear elasticity, thin beams, equilibrium configurations, stationary points