Chouaieb Nadia, Goriely Alain, Maddocks John H
Institut Préparatoire aux Etudes d'Ingénieurs d'El Manar, 2092 El Manar, Tunisia.
Proc Natl Acad Sci U S A. 2006 Jun 20;103(25):9398-403. doi: 10.1073/pnas.0508370103. Epub 2006 Jun 12.
Helices are among the simplest shapes that are observed in the filamentary and molecular structures of nature. The local mechanical properties of such structures are often modeled by a uniform elastic potential energy dependent on bending and twist, which is what we term a rod model. Our first result is to complete the semi-inverse classification, initiated by Kirchhoff, of all infinite, helical equilibria of inextensible, unshearable uniform rods with elastic energies that are a general quadratic function of the flexures and twist. Specifically, we demonstrate that all uniform helical equilibria can be found by means of an explicit planar construction in terms of the intersections of certain circles and hyperbolas. Second, we demonstrate that the same helical centerlines persist as equilibria in the presence of realistic distributed forces modeling nonlocal interactions as those that arise, for example, for charged linear molecules and for filaments of finite thickness exhibiting self-contact. Third, in the absence of any external loading, we demonstrate how to construct explicitly two helical equilibria, precisely one of each handedness, that are the only local energy minimizers subject to a nonconvex constraint of self-avoidance.
螺旋线是在自然界的丝状和分子结构中观察到的最简单形状之一。这类结构的局部力学性质通常由一个依赖于弯曲和扭转的均匀弹性势能来建模，我们将其称为杆模型。我们的第一个结果是完成了由基尔霍夫发起的对不可伸长、不可剪切的均匀杆的所有无限螺旋平衡的半逆分类，这些杆的弹性能量是弯曲和扭转的一般二次函数。具体来说，我们证明了所有均匀螺旋平衡都可以通过某些圆和双曲线的交点，借助显式的平面构造找到。其次，我们证明了在存在模拟非局部相互作用的实际分布力的情况下，相同的螺旋中心线作为平衡态持续存在，例如对于带电线性分子和表现出自接触的有限厚度细丝所产生的非局部相互作用。第三，在没有任何外部载荷的情况下，我们展示了如何明确构造两个螺旋平衡，每种手性恰好一个，它们是受自回避非凸约束的仅有的局部能量极小值。
