Analysis of cone-like singularities in twisted elastic ribbons
B. Audoly & G.H.M. van der Heijden
Twisting a thin elastic ribbon is known to produce a localised deformation
pattern resembling a cone whose tip is located on the edge of the ribbon.
Using the theory of inextensional ribbons, we present a matched asymptotic
analysis of these singularities for ribbons whose width-to-length ratio
w/l is small. An inner layer solution is derived from the
finite-w Wunderlich model and captures the fast, local variations of
the bending and twisting strains in the neighbourhood of the cone-like
region; it is universal up to a load intensity factor. The outer solution
is given by the zero-w Sadowsky model. Based on this analysis, we
propose a new standalone ribbon model that combines the Sadowsky equations
with jump conditions providing a coarse-grained description of cone-like
singularities, and give a self-contained variational derivation of this
model. Applications to the Möbius band and to an end-loaded open ribbon
are presented. Overall, the new model delivers highly accurate
approximations to the solutions of the Wunderlich model in the limit
w ≪ l while avoiding the numerical difficulties associated
with cone-like singularities.
keywords: stress localisation, elastic material, asymptotic analysis, energy
methods
J. Mech. Phys. Solids 171, 105131 (2023)