# Crumpling

In geometry and topology, **crumpling** is the process whereby a sheet of paper or other two-dimensional manifold undergoes disordered deformation to yield a three-dimensional structure comprising a random network of ridges and facets with variable density. The geometry of crumpled structures is the subject of some interest the mathematical community within the discipline of topology.^{[1]} Crumpled paper balls have been studied and found to exhibit surprisingly complex structures with compressive strength resulting from frictional interactions at locally flat facets between folds.^{[2]} The unusually high compressive strength of crumpled structures relative to their density is of interest in the disciplines of materials science and mechanical engineering.

## Significance of crumpling[edit]

The packing of a sheet by crumpling is a complex phenomenon that depends on material parameters and the packing protocol. Thus the crumpling behaviour of foil, paper and poly-membranes differs significantly and can be interpreted on the basis of material foldability.^{[3]} The high compressive strength exhibited by dense crumple formed cellulose paper is of interest towards impact dissipation applications and has been proposed as an approach to utilising waste paper.^{[4]}

From a practical standpoint, crumpled balls of paper are commonly used as toys for domestic cats.

## References[edit]

**^**Cerda, Enrique; Chaieb, Sahraoui; Melo, Francisco; Mahadevan, L (1999). "Conical dislocations in crumpling".*Nature*.**401**: 46–49. Bibcode:1999Natur.401...46C. doi:10.1038/43395.**^**Cambou, Anne Dominique; Narayanan, Menon (2011). "Three-dimensional structure of a sheet crumpled into a ball".*Proceedings of the National Academy of Sciences*.**108**(36): 14741–14745. arXiv:1203.5826. Bibcode:2011PNAS..10814741C. doi:10.1073/pnas.1019192108. PMC 3169141. PMID 21873249.**^**Habibi, M; Bonn, D (2017). "Effect of the material properties on the crumpling of a thin sheet".*Soft matter*.**3**: 4029.**^**Hanaor, D.A.H.; Johnson, E.A. Flores; Wang, S.; Quach, S.; Dela-Torre, K.N.; Gan, Y.; Shen, L. (2017). "Mechanical properties in crumple-formed paper derived materials subjected to compression".*Heilyon*.**3**(6): e00329. doi:10.1016/j.heliyon.2017.e00329.

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