Convergence of equilibria of three-dimensional thin elastic beams

Details Convergence of equilibria of three-dimensional thin elastic beams Convergence of equilibria of three-dimensional thin elastic beams

2006-12-18

arxiv.org/abs/math/0612519v1

Mathematics

Analysis of PDEs

22 pages

Scientific paper

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 Ch^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 M\"uller 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.

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