Starostin E L, van der Heijden G H M
Centre for Nonlinear Dynamics, Department of Civil and Environmental Engineering, University College London, London WC1E 6BT, UK.
Nat Mater. 2007 Aug;6(8):563-7. doi: 10.1038/nmat1929. Epub 2007 Jul 15.
The Möbius strip, obtained by taking a rectangular strip of plastic or paper, twisting one end through 180 degrees, and then joining the ends, is the canonical example of a one-sided surface. Finding its characteristic developable shape has been an open problem ever since its first formulation in refs 1,2. Here we use the invariant variational bicomplex formalism to derive the first equilibrium equations for a wide developable strip undergoing large deformations, thereby giving the first non-trivial demonstration of the potential of this approach. We then formulate the boundary-value problem for the Möbius strip and solve it numerically. Solutions for increasing width show the formation of creases bounding nearly flat triangular regions, a feature also familiar from fabric draping and paper crumpling. This could give new insight into energy localization phenomena in unstretchable sheets, which might help to predict points of onset of tearing. It could also aid our understanding of the relationship between geometry and physical properties of nano- and microscopic Möbius strip structures.
通过取一条塑料或纸条，将一端扭转180度，然后连接两端而得到的莫比乌斯带，是单侧曲面的典型例子。自从它在参考文献1、2中首次被提出以来，找到其特征可展形状一直是一个悬而未决的问题。在这里，我们使用不变变分双复形形式主义来推导一个经历大变形的宽可展带的第一个平衡方程，从而首次非平凡地证明了这种方法的潜力。然后我们为莫比乌斯带制定边值问题并进行数值求解。宽度增加时的解显示出界定几乎平坦三角形区域的折痕的形成，这也是织物悬垂和纸张起皱中常见的特征。这可能会为不可拉伸薄片中的能量局部化现象提供新的见解，这可能有助于预测撕裂起始点。它还可以帮助我们理解纳米和微观莫比乌斯带结构的几何形状与物理性质之间的关系。
