In this paper, we derive the continuous space-time equations of motion of a three-dimensional geometrically exact rod, or the Cosserat rod, incorporating planar cross-sectional deformation. We then adopt the Lie group variational integrator technique to obtain a discrete model of the rod incorporating both rotational motion and cross-sectional deformation as well. The resulting discrete model possesses several desirable features: it ensures volume conservation of the discrete elements by considering cross-sectional deformation through a local dilatation factor, it demonstrates the beneficial properties associated with the variational integrator technique, such as the preservation of the rotational configuration, and energy conservation with a bounded error. An exhaustive set of numerical results under various initial conditions of the rod demonstrates the efficacy of the model in replicating the physics of the system.

翻译：本文推导了考虑平面截面变形的三维几何精确杆（即Cosserat杆）的连续时空运动方程。随后，我们采用李群变分积分器技术，获得了一个同时包含旋转运动与截面变形的杆的离散模型。所得的离散模型具备若干理想特性：通过引入局部膨胀因子考虑截面变形，确保了离散单元的体积守恒；展现了变分积分器技术相关的有益性质，例如旋转构型的保持，以及误差有界的能量守恒。针对杆在不同初始条件下的一系列详尽数值结果，验证了该模型在复现系统物理特性方面的有效性。