A comparative study of crumpling and folding of thin sheets

TL;DR

This paper compares crumpling and folding of thin sheets, revealing their similarities in hierarchical structure, stiffness buildup, and force scaling, despite their apparent differences in randomness and regularity.

Contribution

The study demonstrates that crumpling and folding share underlying geometrical and mechanical principles, allowing simple folding models to describe crumpling behavior.

Findings

Crumpling is hierarchical with increasing layers during compaction.

Crumpling and folding forces follow a power-law relationship with the number of layers.

The dimensionality of the process determines the force scaling exponent.

Abstract

Crumpling and folding of paper are at rst sight very di erent ways of con ning thin sheets in a small volume: the former one is random and stochastic whereas the latest one is regular and deterministic. Nevertheless, certain similarities exist. Crumpling is surprisingly ine cient: a typical crumpled paper ball in a waste-bin consists of as much as 80% air. Similarly, if one folds a sheet of paper repeatedly in two, the necessary force becomes so large that it is impossible to fold it more than 6 or 7 times. Here we show that the sti ness that builds up in the two processes is of the same nature, and therefore simple folding models allow to capture also the main features of crumpling. An original geometrical approach shows that crumpling is hierarchical, just as the repeated folding. For both processes the number of layers increases with the degree of compaction. We nd that for both processes the crumpling force increases as a power law with the number of folded layers, and that the dimensionality of the compaction process (crumpling or folding) controls the exponent of the scaling law between the force and the compaction ratio.