Equilibrium shapes with stress localisation for
inextensible elastic Möbius and other strips
E.L. Starostin & G.H.M. van der Heijden
We formulate the problem of finding equilibrium shapes of a thin inextensible
elastic strip, developing further our previous work on the Möbius strip.
By using the isometric nature of the deformation we reduce the variational
problem to a second-order one-dimensional problem posed on the centreline of
the strip. We derive Euler-Lagrange equations for this problem in
Euler-Poincaré form and formulate boundary-value problems for closed
symmetric one- and two-sided strips. Numerical solutions for the Möbius
strip show a singular point of stress localisation on the edge of the strip,
a generic response of inextensible elastic sheets under torsional strain. By
cutting and pasting operations on the Möbius strip solution, followed by
parameter continuation, we construct equilibrium solutions for strips with
different linking numbers and with multiple points of stress localisation.
Solutions reveal how strips fold into planar or self-contacting shapes as the
length-to-width ratio of the strip is decreased. Our results may be relevant
for curvature effects on physical properties of extremely thin
two-dimensional structures as for instance produced in nanostructured origami.
keywords: Möbius strip, inextensible ribbon, developable surface,
switching point, equilibrium, invariant variational formulation, stress
localisation
Journal of Elasticity 119, 67-112 (2015)