crumpling
{{Short description | Deformation of sheet into complex 3D structure}}
{{Redirect2|Crumpled|Crumple|the deformation feature|Crumple zone|the album by Code Kunst|Crumple (album)}}
In geometry and topology, crumpling is the process whereby a sheet of paper or other two-dimensional manifold undergo 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 of interest to mathematicians studying topology.{{cite journal| last1=Cerda | first1=Enrique | last2=Chaieb |first2=Sahraoui |last3=Melo | first3=Francisco| last4=Mahadevan| first4=L |title=Conical dislocations in crumpling | journal= Nature | year= 1999| volume=401| issue=6748 | doi=10.1038/43395| bibcode=1999Natur.401...46C | pages=46–49| s2cid=4331085 }} 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.{{cite journal |last1=Cambou| first1=Anne Dominique |last2=Narayanan | first2=Menon| title=Three-dimensional structure of a sheet crumpled into a ball.| journal= Proceedings of the National Academy of Sciences |year=2011 | volume=108 |issue=36 |pages=14741–14745|arxiv=1203.5826|bibcode=2011PNAS..10814741C|doi=10.1073/pnas.1019192108|pmc=3169141 | pmid=21873249| doi-access=free }} 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
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.{{cite journal |last1=Habibi| first1=M |last2=Bonn | first2=D| title=Effect of the material properties on the crumpling of a thin sheet.| journal= Soft Matter |year=2017 | volume=3 | issue=22 |pages=4029–4034| doi=10.1039/C6SM02817A | pmid=28512658 | bibcode=2017SMat...13.4029H }} 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.{{cite journal| title=Mechanical properties in crumple-formed paper derived materials subjected to compression| journal= Heliyon|year=2017 | volume=3 |issue=6| pages=e00329 | url= http://www.heliyon.com/article/e00329/ | doi=10.1016/j.heliyon.2017.e00329 | last1 = Hanaor | first1 = D.A.H. | last2 = Johnson | first2 = E.A. Flores | last3 = Wang | first3 = S. | last4 = Quach | first4 = S. | last5 = Dela-Torre | first5 = K.N. | last6 = Gan | first6 = Y. | last7 = Shen | first7 = L.| pmid= 28653042| pmc= 5477149| bibcode= 2017Heliy...300329H| doi-access = free }}
From a practical standpoint, crumpled balls of paper are commonly used as toys for domestic cats.