# Crumpling

Ball of crumpled paper

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

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]

## References

1. ^ Cerda, Enrique; Chaieb, Sahraoui; Melo, Francisco; Mahadevan, L (1999). "Conical dislocations in crumpling". nature. 401. Bibcode:1999Natur.401...46C. doi:10.1038/43395.
2. ^ 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.
3. ^ Habibi, M; Bonn, D (2017). "Effect of the material properties on the crumpling of a thin sheet". Soft matter. 3: 4029.
4. ^ "Mechanical properties in crumple-formed paper derived materials subjected to compression". Heilyon. 3 (6): e00329. 2017.