The equilibrium
shape
of an elastic
developable Möbius strip




Figure: Computed Möbius strips of aspect ratios 5π,
2π, π, 2π/3.


Colour codes the bending
energy density (from violet for almost flat regions to red for highly bent).

The Möbius strip obtained by taking a rectangular strip of
plastic or paper,
twisting one end through 180
degree, and then joining the ends,
is the canonical example of a
onesided 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.
We assume that the material obeys
Hooke's linear law for bending,
then the energy is proportional
to the integral of the nonzero principal curvature squared over the surface of
the strip,
which is 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 in 1930 by M.Sadowsky who
turned it into a 1D variational problem represented in a form that is invariant
under Euclidean motions.
Later
W.Wunderlich generalised this formulation to a strip of finite width,
but the problem has remained open 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 finitewidth
developable strip thereby giving the first nontrivial demonstration of the
potential of this approach. The boundaryvalue problem for the Möbius strip offers a fitting example for application of
these equations.
Numerical
solutions for increasing widthtolength
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.






Figure: Computed Möbius strip of aspect ratio 5π.


Colour shows the bending energy
density (from violet for almost flat regions to red for highly bent).







Figure: Computed Möbius strip of aspect ratios 5π,
2π, π, 2π/3.


Colour shows the bending
energy density (from violet for almost flat regions to red for highly bent).
