Forceless folding of thin annular strips
E.L. Starostin & G.H.M. van der Heijden
Thin strips or sheets with in-plane curvature have a natural tendency to
adopt highly symmetric shapes when forced into closed structures and to
spontaneously fold into compact multi-covered configurations under feed-in
of more length or change of intrinsic curvature. This disposition is
exploited in nature as well as in the design of everyday items such as
foldable containers. We formulate boundary-value problems (for an ODE) for
symmetric equilibrium solutions of unstretchable circular annular strips
and present sequences of numerical solutions that mimic different folding
modes. Because of the high-order symmetry, closed solutions cannot have an
internal force, i.e., the strips are forceless. We consider both wide and
narrow (strictly zero-width) strips. Narrow strips cannot have inflections,
but wide strips can be either inflectional or non-inflectional. Inflectional
solutions are found to feature stress localisations, with divergent strain
energy density, on the edge of the strip at inflections of the surface.
'Regular' folding gives these singularities on the inside of the annulus,
while 'inverted' folding gives them predominantly on the outside of the
annulus. No new inflections are created in the folding process as more
length is inserted. We end with a discussion of an intriguing apparent
connection with a deep result on the topology of curves on surfaces.
keywords: thin strips, elastic equilibria, developable surface, annulus,
stress localisation, singularity
J. Mech. Phys. Solids 169, 105054 (2022)