Crumpling of Curved Sheets: Generalizing Foeppl-von Karman

TL;DR

This paper extends the F"{o}ppl-von Kármán equations to initially curved sheets, providing a geometric shell theory that applies to both flat and strongly curved structures, including biological membranes.

Contribution

It introduces a generalized formulation of the F"{o}ppl-von Kármán equations for precurved shells, bridging a gap in shell theory and biological membrane modeling.

Findings

Derivation of generalized equations for curved sheets.

Reduction to classic equations in the flat limit.

Application to biological membranes and crumpling phenomena.

Abstract

We generalize the F\"{o}ppl-von K\'arm\'an equations to an initially precurved sheet and present the underlying derivation. A geometrically computed moment of strain replaces the notion of bending moment and results in a geometric formulation of the theory of shells. As the curvature approaches zero, i.e., the sheet becomes flat, the new equations reduce to the classic F\"{o}ppl-von K\'arm\'an ones. The present theory solves the long-standing problem of formulating these equations for an a priori curved shell and applies, for instance, both to shell theory and to strongly curved biomembranes of cells as closed surfaces, exhibiting crumpling as the membrane thickness goes to zero.