How paper folds: bending with local constraints

TL;DR

This paper introduces a variational framework to analyze how surfaces, like paper, bend under local geometric constraints, linking stress and curvature to understand folding patterns.

Contribution

It develops a novel variational approach incorporating local constraints and identifies the associated stress-curvature coupling in folded sheets.

Findings

Framework successfully models folding patterns

Identifies stress as a conserved quantity

Applies to deformation of flat sheets into cones

Abstract

A variational framework is introduced to describe how a surface bends when it is subject to local constraints on its geometry. This framework is applied to describe the patterns of a folded sheet of paper. The unstretchability of paper implies a constraint on the surface metric; bending is penalized by an energy quadratic in mean curvature. The local Lagrange multipliers enforcing the constraint are identified with a conserved tangential stress that couples to the extrinsic curvature of the sheet. The framework is illustrated by examining the deformation of a flat sheet into a generalized cone.