Thin-walled beams with a cross-section of arbitrary geometry: derivation of linear theories starting from 3D nonlinear elasticity

TL;DR

This paper rigorously derives linear models for thin-walled beams with arbitrary cross-sectional geometry from 3D nonlinear elasticity, considering various scaling regimes of the cross-section's diameter and thickness.

Contribution

It provides a systematic derivation of lower-dimensional linear theories for thin-walled beams with arbitrary cross-sections from 3D nonlinear elasticity, depending on geometric scaling.

Findings

Different linearized models are obtained based on the relative size of cross-section diameter and thickness.

The analysis covers various scaling regimes of the elastic energy.

The work extends classical beam theories to more general geometries and nonlinear elasticity.

Abstract

The subject of this paper is the rigorous derivation of lower dimensional models for a nonlinearly elastic thin-walled beam whose cross-section is given by a thin tubular neighbourhood of a smooth curve. Denoting by h and {\delta}_h, respectively, the diameter and the thickness of the cross-section, we analyse the case where the scaling factor of the elastic energy is of order {\epsilon}_h^2, with {\epsilon}_h/{\delta}_h^2 \rightarrow l \in [0, +\infty). Different linearized models are deduced according to the relative order of magnitude of {\delta}_h with respect to h.