Convergence of equilibria of three-dimensional thin elastic beams

2006/12/18 by Maria Giovanna Mora, Stefan Müller, Mora, Maria Giovanna +1 · 2 citations
Computer Science · Engineering · Mathematics · #74B20 #74G10 #74K10 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Contact Mechanics and Variational Inequalities #Elasticity and Material Modeling #FOS: Mathematics #math.AP #msc:74B20 #msc:74G10 #msc:74K10

paper · pdf · doi:10.48550/arxiv.math/0612519

22 pages

arxiv created 2006/12/18 · openalex publication_date 2006/12/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Ch², converge to stationary points of the Gamma-limit of E^h/h². 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.

Citations

• A theorem on geometric rigidity and the derivation of nonlinear plate theory from three‐dimensional elasticity
• A Hierarchy of Plate Models Derived from Nonlinear Elasticity by Gamma-Convergence

Cited by

• Shape-Transition in Non-Euclidean Ribbons
• Dimension reduction for time-dependent von Kármán rods

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