The cross-sectional warping problem for hyperelastic beams: An efficient formulation in Voigt notation

TL;DR

This paper introduces an efficient Voigt notation-based formulation for the cross-sectional warping problem in hyperelastic beams, enabling nonlinear analysis beyond traditional small strain assumptions.

Contribution

It reinterprets the warping problem for hyperelastic beams using Green-Lagrange and Piola-Kirchhoff tensors, facilitating numerical implementation and reproducibility.

Findings

Validated the formulation with numerical examples

Derived sensitivities of warping displacement

Computed effective beam stiffness

Abstract

Beam theory has traditionally been restricted to small elastic strains and rigid cross-sections. Relaxing these assumptions within closed-form analytical frameworks remains challenging. In contrast, the cross-sectional warping problem provides a computational approach that enables the derivation of general, nonlinear constitutive relations for beam models, thereby overcoming both limitations. In this work, we reinterpret the cross-sectional warping problem for hyperelastic beams and propose a fully material formulation in terms of the Green-Lagrange strain and the second Piola-Kirchhoff stress tensors. Owing to the symmetry of these tensors, the formulation can be expressed efficiently in Voigt notation and is thus particularly well-suited for straightforward numerical implementation. We demonstrate the validity of this alternative formulation in numerical examples, including the computation of the effective beam stiffness, for which we derive the sensitivities of the warping displacement. To promote reproducibility, we accompany this article with an open-access repository containing the isogeometric finite element implementation and all numerical examples presented herein, enabling other researchers to readily reproduce and build upon our results.