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MiS Preprint

Convergence of equilibria of three-dimensional thin elastic beams

Maria Giovanna Mora and Stefan Müller


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.

MSC Codes:
74K10, 74B20, 74G10
beams, nonlinear elasticity, dimension reduction, stationary points

Related publications

2008 Repository Open Access
Maria Giovanna Mora and Stefan Müller

Convergence of equilibria of three-dimensional thin elastic beams

In: Proceedings of the Royal Society of Edinburgh / A, 138 (2008) 4, pp. 873-896