Editing Poisson's ratio (section)

==Origin==
Poisson's ratio is a measure of the Poisson effect, the phenomenon in which a material tends to expand in directions perpendicular to the direction of compression. Conversely, if the material is stretched rather than compressed, it usually tends to contract in the directions transverse to the direction of stretching. It is a common observation when a rubber band is stretched, it becomes noticeably thinner. Again, the Poisson ratio will be the ratio of relative contraction to relative expansion and will have the same value as above. In certain rare cases,<ref name=lakes>{{cite journal|last1=Lakes |first1=R. |last2=Wojciechowski |first2=K. W. |date=2008 |title=Negative compressibility, negative Poisson's ratio, and stability |journal=Physica Status Solidi B |volume=245 |issue=3 |pages=545–551|doi=10.1002/pssb.200777708 |bibcode=2008PSSBR.245..545L }}</ref> a material will actually shrink in the transverse direction when compressed (or expand when stretched) which will yield a negative value of the Poisson ratio.

The Poisson's ratio of a stable, [[isotropy#Materials science|isotropic]], linear [[Elasticity (physics)|elastic]] material must be between −1.0 and +0.5 because of the requirement for [[Young's modulus]], the [[shear modulus]] and [[bulk modulus]] to have positive values.<ref>{{cite journal | last1 = Gercek | first1 = H. | date = January 2007 | title = Poisson's ratio values for rocks | journal = International Journal of Rock Mechanics and Mining Sciences | volume = 44 | issue = 1| pages = 1–13 | doi=10.1016/j.ijrmms.2006.04.011| bibcode = 2007IJRMM..44....1G }}</ref> Most materials have Poisson's ratio values ranging between 0.0 and 0.5. A perfectly incompressible isotropic material deformed elastically at small strains would have a Poisson's ratio of exactly 0.5. Most steels and rigid polymers when used within their design limits (before [[Yield (engineering)|yield]]) exhibit values of about 0.3, increasing to 0.5 for post-yield deformation which occurs largely at constant volume.<ref>{{cite book|last=Park |first=R. J. T. |title=Seismic Performance of Steel-Encased Concrete Piles}}{{full citation needed|date=April 2024}}</ref> Rubber has a Poisson ratio of nearly 0.5. Cork's Poisson ratio is close to 0, showing very little lateral expansion when compressed. Glass is between 0.18 and 0.30. Some materials, e.g. some polymer foams, origami folds,<ref>{{Cite book| url=http://www.markschenk.com/research/files/PhD%20thesis%20-%20Mark%20Schenk.pdf | title=Folded Shell Structures, PhD Thesis |last=Mark|first=Schenk |publisher=University of Cambridge, Clare College |year=2011}}</ref><ref>{{Cite journal | last1=Wei|first1=Z. Y.| last2=Guo|first2=Z. V.| last3=Dudte|first3=L.| last4=Liang|first4=H. Y.| last5=Mahadevan|first5=L.| date=2013-05-21| title=Geometric Mechanics of Periodic Pleated Origami | url=https://www.seas.harvard.edu/softmat/downloads/2013-09.pdf |journal=Physical Review Letters|volume=110|issue=21| pages=215501| doi=10.1103/PhysRevLett.110.215501 | pmid=23745895| arxiv=1211.6396 | bibcode=2013PhRvL.110u5501W | s2cid=9145953}}</ref> and certain cells can exhibit negative Poisson's ratio, and are referred to as [[auxetics|auxetic materials]]. If these auxetic materials are stretched in one direction, they become thicker in the perpendicular direction. In contrast, some [[Anisotropy|anisotropic]] materials, such as [[carbon nanotube]]s, zigzag-based folded sheet materials,<ref name=":0">{{Cite journal| last1=Eidini|first1=Maryam| last2=Paulino|first2=Glaucio H.| date=2015| title=Unraveling metamaterial properties in zigzag-base folded sheets| journal=Science Advances|volume=1|issue=8|pages=e1500224| arxiv=1502.05977| bibcode=2015SciA....1E0224E| doi=10.1126/sciadv.1500224| issn=2375-2548| pmid=26601253| pmc=4643767}}</ref><ref>{{Cite journal | last=Eidini | first=Maryam | title=Zigzag-base folded sheet cellular mechanical metamaterials | journal=Extreme Mechanics Letters|volume=6 | pages=96–102 |doi=10.1016/j.eml.2015.12.006 |arxiv=1509.08104 |year=2016 | bibcode=2016ExML....6...96E |s2cid=118424595}}</ref> and honeycomb auxetic metamaterials<ref>{{Cite journal |last1=Mousanezhad|first1=Davood |last2=Babaee|first2=Sahab |last3=Ebrahimi|first3=Hamid |last4=Ghosh|first4=Ranajay |last5=Hamouda|first5=Abdelmagid Salem |last6=Bertoldi|first6=Katia |last7=Vaziri|first7=Ashkan |date=2015-12-16 |title=Hierarchical honeycomb auxetic metamaterials |journal=Scientific Reports|volume=5 |doi=10.1038/srep18306 |issn=2045-2322 |pmc=4680941|pmid=26670417 |page=18306 |bibcode=2015NatSR...518306M}}</ref> to name a few, can exhibit one or more Poisson's ratios above 0.5 in certain directions.

Assuming that the material is stretched or compressed in only one direction (the {{mvar|x}} axis in the diagram below):
:<math>\nu = -\frac{d\varepsilon_\mathrm{trans}}{d\varepsilon_\mathrm{axial}} = -\frac{d\varepsilon_\mathrm{y}}{d\varepsilon_\mathrm{x}}= -\frac{d\varepsilon_\mathrm{z}}{d\varepsilon_\mathrm{x}} </math>
where
*{{mvar|ν}} is the resulting Poisson's ratio,
*{{math|''ε''<sub>trans</sub>}} is transverse strain
*{{math|''ε''<sub>axial</sub>}} is axial strain
and positive strain indicates extension and negative strain indicates contraction.