Crumpling of Curved Sheets: Generalizing Foeppl-von Karman

Abstract

We generalize the Föppl-von Kármán 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 Föppl-von Kármán 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.