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 nano-scale force probe devices.
Our model generalizes the Sadowsky-Wunderlich 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 3-space. The elastic energy is described by the Kirchhoff-Love 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 Euclidean-invariant form.
The Euler-Lagrange 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 non-uniform coiling along its end-to-end axis which may
relate to the tension-induced straightening transition observed in experiments with cholesterol ribbons.
Helical perversions also may occur depending on the boundary conditions.
The resulting force-extension 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.