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

2026-04-14

AI Summary

This study addresses the limitations of classical beam theories—rooted in small-strain assumptions and rigid cross-section constraints—in capturing large deformations and cross-sectional warping in hyperelastic materials. To overcome these challenges, the authors develop a fully material-consistent constitutive framework based on Green-Lagrange strain and the second Piola-Kirchhoff stress. For the first time, they incorporate the Voigt representation of symmetric tensors into the modeling of hyperelastic beam warping, substantially enhancing computational efficiency and enabling direct numerical implementation. By integrating isogeometric finite element methods, the formulation further derives the sensitivity of effective stiffness with respect to warping displacements. The proposed approach accurately predicts the effective stiffness of hyperelastic beams, and all source code and numerical examples are made publicly available to ensure full reproducibility.

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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.

Problem

• cross-sectional warping
• hyperelastic beams
• nonlinear constitutive relations
• Green-Lagrange strain
• beam theory

Innovation

• hyperelastic beams
• cross-sectional warping
• Voigt notation
• Green-Lagrange strain
• isogeometric finite element