University of Cambridge > > Engineering Department Structures Research Seminars > New equations for inextensible sheets with applications to Moebius strips and helical nanoribbons

New equations for inextensible sheets with applications to Moebius strips and helical nanoribbons

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We derive a new set of geometrically-exact equilibrium equations for the deformation of thin inextensible strips of finite width. The equations are the Euler-Lagrange equations for a geometrical variational problem with a functional in terms of the curvature and torsion of the strip’s axial curve as well as their derivatives with respect to arclength. The equations are used to solve the long-standing problem of finding the characteristic shape of a material Moebius strip. Solutions for increasing width-to-length ratio show the formation of creases bounding nearly flat triangular regions, a feature also familiar from fabric draping and paper crumpling. This suggests that our approach could give new insight into energy localisation phenomena in unstretchable elastic sheets, which for instance could help to predict points of onset of tearing. The technique for deriving equilibrium equations can be generalised to intrinsically curved sheets (shells). In the second part of the talk we will apply this to study the force-extension behaviour of helical ribbons. A complete analytical study of stretched/compressed exact helical solutions can be carried out. Unlike previous rod models our strip model predicts hysteresis behaviour for low-pitch ribbons of arbitrary material properties. Associated with it is a first-order transition between two different helical states, a phenomenon observed in experiments with cholesterol ribbons. Numerical solutions for non-helical solutions reveal a new non-uniform uncoiling scenario in which a ribbon of very low pitch shears under tension and successively releases a sequence of almost planar loops. Our results may be relevant for nanoscale devices such as force probes.

This talk is part of the Engineering Department Structures Research Seminars series.

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