Convergence of equilibria of three-dimensional thin elastic beams

Maria Giovanna Mora and Stefan Müller

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Submission date: 10. Nov. 2006
Pages: 24
published in: Proceedings of the Royal Society of Edinburgh / A, 138 (2008) 4, p. 873-896 
DOI number (of the published article): 10.1017/S0308210506001120
Bibtex
MSC-Numbers: 74K10, 74B20, 74G10
Keywords and phrases: beams, nonlinear elasticity, dimension reduction, stationary points
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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 , whose energies (per unit cross-section) are bounded by , converge to stationary points of the -limit of . 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.