*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

**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 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

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