Stretching
inextensible
helical
ribbons


We present a novel elastic
model for statics of large isometric deformations of a helical ribbon. Helical ribbons are common in
nature and, in particular, are promising as parts of nanoscale force probe
devices. Our model generalizes the
SadowskyWunderlich description of thin developable strips by allowing for an
intrinsic curvature. We present a novel elastic model for statics of large
isometric deformations of a helical ribbon. 
In
its relaxed state, the ribbon is assumed to belong to a cylindrical surface.
When deformed, its surface remains to be isometrically embedded into 3space.
The elastic energy is described by the KirchhoffLove shell model.
For
a strip of given width, the free energy can be reduced to a 1D functional in
terms of the curvature and torsion of the centreline. Finding equilibrium
configurations implies solution of the
variational problem which is expressed in Euclideaninvariant form.
The
EulerLagrange equations may be directly obtained in invariant form, too.
The equilibrium shape of the ribbon
is described by a system of ODEs which is solved for specific boundary
conditions.
We carry out an analytical study
of stretched exact helical solutions.
We obtain an explicit expression
for the applied force as a function of the helix axis length
and we analyse the dependence of
this function on the geometric parameters of the unperturbed helix and on the
Poisson ratio.
In
the numerical analysis, we concentrate on the limiting case of an
infinitesimally narrow ribbon.
We
solve the boundary value problem for various sets of conditions at the ends
corresponding to various physical realizations
of
the external force/torque. We also investigate the influence of parameters
including the number of turns in the relaxed helix.
As
a helical ribbon is stretched, it unwinds and we observe nonuniform coiling
along its endtoend axis which may
relate
to the tensioninduced straightening transition observed in experiments with
cholesterol ribbons.
Helical
perversions also may occur depending on the boundary conditions.
The resulting forceextension
diagram reveals multiple local extrema that previously were only known in the
helical elastic thin rod models for
large torsional rigidity. Our elastic ribbon model predicts the multiple
hysteresis transitions for material with arbitrary Poisson ratio and for a
range of the spontaneous helical pitch angle depending on this ratio.