Calculus of Variations and Geometric Measure Theory

S. Conti - F. Maggi

Confining thin elastic sheets and folding paper

created by maggi on 14 Jul 2005
modified on 08 Jun 2010


Accepted Paper

Inserted: 14 jul 2005
Last Updated: 8 jun 2010

Journal: Archive for Rational Mechanics and Analysis
Year: 2005

Download at: http:/analysis.math.uni-duisburg.depublicationspreprintsContiMaggi.html


Crumpling a sheet of paper leads to the formation of complex folding patterns over several length scales. This can be understood on the basis of the interplay of a nonconvex elastic energy, which favors locally isometric deformations, and a small singular perturbation, which penalizes high curvature.

Based on three-dimensional nonlinear elasticity and by using a combination of explicit constructions and general results from differential geometry, we prove that, in agreement with previous heuristic results in the physics literature, the total energy per unit thickness of such folding patterns scales at most as the thickness of the sheet to the power 53. For the case of a ``single fold'' we also obtain a corresponding lower bound.

Keywords: nonlinear elasticity, Gamma convergence, isometric embeddings, thin elastic bodies