Dipoles in thin sheets

Abstract

A flat elastic sheet may contain pointlike conical singularities that carry a metrical "charge" of Gaussian curvature. Adding such elementary defects to a sheet allows one to make many shapes, in a manner broadly analogous to the familiar multipole construction in electrostatics. However, here the underlying field theory is non-linear, and superposition of intrinsic defects is non-trivial as it must respect the immersion of the resulting surface in three dimensions. We consider a "charge-neutral" dipole composed of two conical singularities of opposite sign. Unlike the relatively simple electrostatic case, here there are two distinct stable minima and an infinity of unstable equilibria. We determine the shapes of the minima and evaluate their energies in the thin-sheet regime where bending dominates over stretching. Our predictions are in surprisingly good agreement with experiments on paper sheets.

20 pages, 5 figures, 2 tables

References in corpus (7)

• Morphogenesis of growing soft tissues
• Conical defects in growing sheets
• Confinement of semiflexible polymers
• Self-contact and instabilities in the anisotropic growth of elastic membranes
• Curvature condensation and bifurcation in an elastic shell
• How paper folds: bending with local constraints
• Conical instabilities on paper

Cited by in corpus (10)

• On Local Kirigami Mechanics I: Isometric Conical Solutions
• The emergence of torsion in the continuum limit of distributed edge-dislocations
• Cutting holes in bistable folds
• On local kirigami mechanics II: Stretchable creased solutions
• Elasticity in Curved Topographies: Exact Theories and Linear Approximations
• Confined disclinations: exterior vs material constraints in developable thin elastic sheets
• Nonuniform growth and topological defects in the shaping of elastic sheets
• Bistability and equilibria of creased annular sheets and strips
• Jump conditions for strings and sheets from an action principle
• Elasticity and Stability of Shape Changing Structures