Closed Unstretchable Knotless Ribbons and the Wunderlich Functional

Abstract

In 1962, Wunderlich published the article “On a developable Möbius band,” in which he attempted to determine the equilibrium shape of a free standing Möbius band. In line with Sadowsky’s pioneering works on Möbius bands of infinitesimal width, Wunderlich used an energy minimization principle, which asserts that the equilibrium shape of the Möbius band has the lowest bending energy among all possible shapes of the band. By using the developability of the band, Wunderlich reduced the bending energy from a surface integral to a line integral without assuming that the width of the band is small. Although Wunderlich did not completely succeed in determining the equilibrium shape of the Möbius band, his dimensionally reduced energy integral is arguably one of the most important developments in the field. In this work, we provide a rigorous justification of the validity of the Wunderlich integral and fully formulate the energy minimization problem associated with finding the equilibrium shapes of closed bands, including both orientable and nonorientable bands with arbitrary number of twists. This includes characterizing the function space of the energy functional, dealing with the isometry and local injectivity constraints, and deriving the Euler–Lagrange equations. Special attention is given to connecting edge conditions, regularity properties of the deformed bands, determination of the parameter space needed to ensure that the deformation is surjective, reduction in isometry constraints, and deriving matching conditions and jump conditions associated with the Euler–Lagrange equations.

Nonrigidity of flat ribbons

“…This work joins several other recent studies on ribbons; see e.g. [2,7,8,21]. In particular, the problem of constructing flat surfaces along a given curve has also been considered in [12,14,18,28]; interesting applications of Sadowsky's energy formula can be found in [3,4,11,23].…”

Isometries Between Given Surfaces and the Isometric Deformation of a Single Unstretchable Material Surface

“…However, to determine the optimal form the functional cannot be extremized over the class of rectifying developable surfaces as parametrized in (26). One acceptable approach, as Seguin, Chen and Fried [5] have noted, is to extremize an augmented functional that incorporates the constraints that ensure that the deformation is isometric through the introduction of suitable reactions in the form of Lagrange multiplier fields defined on the midline of the band. Of course, the appropriate conditions of smoothness at the connection of the two ends must be stated when defining the class of admissible variations that are used for determining the governing Euler-Lagrange equations -equations which ultimately will provide the field of generators.…”

“…In particular, any boundary condition which involves bending one of the short edges cannot be accommodated. While Wunderlich's dimensional reduction of the bending energy is correct for closed, unknotted ribbons without self intersection, independent of their orientation, as Seguin, Chen and Fried [5] have explained, the variational problem associated with minimum bending energy for such a ribbon must be carried out over an appropriate collection of competitors that differs from the class of rectifying developables, that class being the collection of isometric deformations ỹ defined on the reference region D. This can be achieved by introducing suitable Lagrange multipliers to account for the constraint of unstretchability. The main oversight in the Wunderlich [8,9] approach is that no reference configuration for an undistorted material surface is introduced so as to locate and distinguish the particles of the ribbon that is deformed and thereby properly track the deformation of the ribbon.…”

Construction of Unknotted and Knotted Symmetric Developable Bands

“…Guven et al [6] explored the influence of the constraint of isometry on the boundary conditions that apply along the free edge of an unstretchable surface. Seguin et al [7] extended Wunderlich's [4,5] work to account for the constraints needed to ensure that a solution to the dimensionally reduced variational problem constitutes an isometric and locally injective deformation from a rectangular strip to an unknotted Möbius band, allowing for any number of half twists, and derived the corresponding Euler-Lagrange equations and jump conditions. Starostin and van der Heijden [8] were the first to develop a numerical method for minimizing the Wunderlich [4,5] functional and applied that method to determine the shape of a Möbius band with one half twist.…”