Forceless folding of thin annular strips

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.