Borum Andy, Bretl Timothy
Department of Mathematics, Cornell University, Ithaca, New York 14853, USA.
Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA.
Phys Rev E. 2020 Aug;102(2-1):023004. doi: 10.1103/PhysRevE.102.023004.
It has long been known that every configuration of a planar elastic rod with clamped ends satisfies the property that if its centerline has constant nonzero curvature, then it is in stable equilibrium regardless of its length. In this paper, we show that for a certain class of nonplanar elastic rods, no configuration satisfies this property. In particular, using results from optimal control theory, we show that every configuration of an inextensible, unshearable, isotropic, and uniform Kirchhoff rod with clamped ends that has a helical centerline with constant nonzero curvature becomes unstable at a finite length. We also derive coordinates for computing this critical length that are independent of the rod's bending and torsional stiffness. Finally, we derive a scaling relationship between the length at which a helical rod becomes unstable and the rod's curvature, torsion, and twist. In a companion paper, these results are used to compute the set of all stable rods with helical centerlines.
长期以来，人们都知道，两端固定的平面弹性杆的每一种构型都具有这样的性质：如果其中心线具有恒定的非零曲率，那么无论其长度如何，它都处于稳定平衡状态。在本文中，我们表明，对于某一类非平面弹性杆，不存在满足该性质的构型。具体而言，利用最优控制理论的结果，我们表明，每一个两端固定、不可伸长、不可剪切、各向同性且均匀的基尔霍夫杆的构型，其具有恒定非零曲率的螺旋中心线在有限长度时会变得不稳定。我们还推导了用于计算这个临界长度的坐标，这些坐标与杆的弯曲和扭转刚度无关。最后，我们推导了螺旋杆变得不稳定时的长度与杆的曲率、扭转和扭曲之间的比例关系。在一篇配套论文中，这些结果被用于计算所有具有螺旋中心线的稳定杆的集合。
