We consider the problem of producing a ruled Möbius band by subjecting an unstretchable, homogeneous, isotropic, elastic material surface material surface in a circular helicoidal reference configuration to a deformation that is isometric and chirality preserving. We find that such a Möbius band is completely determined by the unit binormal of the Frenet frame of its midline, which must be a geodesic and must have uniform torsion inversely proportional to the pitch of the helicoidal reference configuration. Granted that the energy density of the material surface depends quadratically on the mean curvature of its deformed configuration, we show that the total energy stored in producing a ruled Möbius band as described reduces, in closed form and without approximation, to an integral over the midline of the Möbius band. We formulate and numerically solve a constrained variational problem for finding relative minima of the dimensionally reduced bending energy and construct corresponding stable Möbius bands. The only input parameter entering our variational problem is the number \(\nu \) of turns in a helicoidal reference configuration. We only find solutions if \(\nu \) exceeds a certain threshold, which we compute to machine precision. Above that threshold, an interplay between the operative constraints leads to a multiplicity of coexisting stable solutions with \(n\ge 3\) half twists. For each \(n\ge 3\), we construct an energetically optimal Möbius band which exhibits \(n\)-fold rotational symmetry. All other energy minima yield Möbius bands which lack symmetry. To our knowledge, this study contains the first examples of stable Möbius bands produced by isometrically deforming reference configurations that are not flat.

我们考虑通过使圆形螺旋参考配置中的不可拉伸、均匀、各向同性、弹性材料表面材料表面受到等距和手性保持的变形来产生规则莫比乌斯带的问题。我们发现这样的 Möbius 带完全由其中线 Frenet 框架的单位副法线决定，它必须是测地线并且必须具有与螺旋参考配置的螺距成反比的均匀扭转。假定材料表面的能量密度以二次方方式取决于其变形配置的平均曲率，我们表明，在产生所描述的莫比乌斯带中存储的总能量以封闭形式且没有近似地减少到中线上的积分莫比乌斯带。我们制定并数值求解约束变分问题，以找到尺寸减小的弯曲能量的相对最小值，并构建相应的稳定莫比乌斯带。进入变分问题的唯一输入参数是数字\(\nu \)圈在螺旋参考配置中。我们只会在\(\nu \)超过某个阈值时找到解决方案，我们计算机器精度。在该阈值之上，操作约束之间的相互作用导致具有\(n\ge 3\)半扭曲的多种共存稳定解决方案。对于每个\(n\ge 3\)，我们构建了一个能量最优的 Möbius 带，它表现出\(n\)倍旋转对称性。所有其他能量最小值产生缺乏对称性的莫比乌斯带。据我们所知，这项研究包含了由不平坦的等距变形参考配置产生的稳定莫比乌斯带的第一个例子。