In this work, we present the mathematical formulation and the numerical implementation of a new model for initially straight, transversely isotropic rods. By adopting a configuration space that intrinsically avoids shear deformations and by systemically neglecting the energetic contribution due to torsion, the proposed model admits an unconstrained variational statement. Moreover, as the natural state of the rod is the trivial one and the mechanical properties are homogeneous on the cross section, the need for pull-back and push-forward operations in the formulation is totally circumvented. These features impose, however, some smoothness requirements on the stored energy function that need to be carefully considered when adopting general constitutive models. In addition to introducing the rod model, we propose a spatial discretization with the finite element method, and a time integration with a hybrid, implicit scheme. To illustrate the favorable features of the new model, we provide results corresponding to numerical simulations for plane and three-dimensional problems that are investigated in the static and dynamic settings. Finally, and to put the presented ideas in a suitable context, we compare solutions obtained with the new model against a rod model that allows for torsion and shear.

在这项工作中，我们提出了初始直、横向各向同性棒的新模型的数学公式和数值实现。通过采用本质上避免剪切变形的配置空间并通过系统地忽略由于扭转引起的能量贡献，所提出的模型承认不受约束的变分陈述。此外，由于棒的自然状态是微不足道的，并且横截面的机械性能是均匀的，因此完全避免了配方中回拉和推进操作的需要。然而，这些特征对存储能量函数提出了一些平滑要求，在采用一般本构模型时需要仔细考虑这些要求。除了介绍杆模型，我们建议使用有限元方法进行空间离散化，并使用混合隐式方案进行时间积分。为了说明新模型的有利特征，我们提供了与在静态和动态环境中研究的平面和三维问题的数值模拟相对应的结果。最后，为了将所提出的想法放在合适的上下文中，我们将使用新模型获得的解决方案与允许扭转和剪切的杆模型进行比较。