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.

翻译：梁理论传统上受限于小弹性应变和刚性横截面。在封闭解析框架内放松这些假设仍然具有挑战性。相比之下，横截面翘曲问题提供了一种计算方法，能够推导出梁模型的通用非线性本构关系，从而同时克服了这两个局限性。在本工作中，我们重新阐释了超弹性梁的横截面翘曲问题，并基于Green-Lagrange应变和第二Piola-Kirchhoff应力张量提出了一种完全的材料公式。由于这些张量的对称性，该公式可以高效地表达为Voigt记法，因此特别适合直接进行数值实现。我们通过数值算例验证了该替代公式的有效性，包括有效梁刚度的计算，并为此推导了翘曲位移的灵敏度。为促进可复现性，本文附有一个开源代码库，其中包含等几何有限元实现及本文所述的所有数值算例，使其他研究者能够轻松复现并拓展我们的结果。