The shape of a Möbius strip
E.L. Starostin & G.H.M. van der Heijden
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. As simple experimentation
shows, a physical Möbius strip, when left to itself, adopts a
characteristic shape independent of the type of material (sufficiently
stiff for gravity to be ignorable). This shape is well described by a
developable surface that minimises the deformation energy, which is entirely
due to bending. If we assume that the material obeys Hooke's linear law for
bending then the energy is proportional to the integral of the non-zero
principal curvature squared over the surface of the strip (taken to be an
isometric embedding of a rectangle into 3D space).
The problem of finding the equilibrium shape of a narrow Möbius strip
was first formulated by M. Sadowsky in 1930 who turned it into a 1D
variational problem expressed in a form that is invariant under Euclidean
motions. Later W. Wunderlich generalised this formulation to a strip of
finite width. However, deriving the (Euler-Lagrange) equilibrium equations
for a finite-width Möbius strip has remained an open problem, although
geometrical constructions of developable Möbius strips have appeared.
We apply an invariant geometrical approach based on the variational
bicomplex formalism to derive the first equilibrium equations for a
finite-width developable strip, 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-to-length ratio show the formation of creases
bounding nearly flat triangular regions, a feature also familiar from fabric
draping and paper crumpling. This suggests that our approach could give new
insight into energy localisation phenomena in unstretchable elastic sheets,
which for instance could help to predict points of onset of tearing.
Nature Materials 6, 563-567 (2007)