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Convergence of equilibria of three-dimensional thin elastic beams. (English) Zbl 1142.74022

Summary: A convergence result is proved for the equilibrium configurations of a three-dimensional thin elastic beam, as the diameter \(h\) of the cross-section tends 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 \(\varGamma\)-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 G. Friesecke, R. D. James and S. Müller [Commun. Pure Appl. Math. 55, No. 11, 1461–1506 (2002; Zbl 1021.74024)] 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:

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics
74G65 Energy minimization in equilibrium problems in solid mechanics
35Q72 Other PDE from mechanics (MSC2000)

Citations:

Zbl 1021.74024
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