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Poisson's ratio
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=== Volumetric change === The relative change of volume {{math|{{sfrac|Δ''V''|''V''}}}} of a cube due to the stretch of the material can now be calculated. Since {{math|''V'' {{=}} ''L''<sup>3</sup>}} and :<math>V + \Delta V = (L + \Delta L)\left(L + \Delta L'\right)^2</math> one can derive :<math>\frac{\Delta V}{V} = \left(1 + \frac{\Delta L}{L} \right)\left(1 + \frac{\Delta L'}{L} \right)^2 - 1</math> Using the above derived relationship between {{math|Δ''L''}} and {{math|Δ''L''′}}: :<math>\frac {\Delta V} {V} = \left(1 + \frac{\Delta L}{L} \right)^{1 - 2\nu} - 1</math> and for very small values of {{math|Δ''L''}} and {{math|Δ''L''′}}, the first-order approximation yields: :<math>\frac {\Delta V} {V} \approx (1-2\nu)\frac{\Delta L}{L}</math> For isotropic materials we can use [[Lamé parameters|Lamé's relation]]<ref>{{Cite journal|last1=Mott |first1=P. H. |last2=Roland |first2=C. M. |title=Limits to Poisson's ratio in isotropic materials—general result for arbitrary deformation |journal=Physica Scripta |publisher=Chemistry Division, Naval Research Laboratory |date=3 April 2012|volume=87 |issue=5 |page=055404 |doi=10.1088/0031-8949/87/05/055404 |arxiv=1204.3859 |s2cid=55920779}}</ref> :<math>\nu \approx \frac{1}{2} - \frac{E}{6K}</math> where {{mvar|K}} is [[bulk modulus]] and {{mvar|E}} is [[Young's modulus]].
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