Poisson's ratio
/ pwah-SOHN /
Poisson's ratio captures a fact everyone has felt: stretch something and it usually gets thinner; squash it and it bulges out sideways. It is the single number that says how much a material contracts crosswise when you stretch it lengthwise. Pull a rubber band and it necks in noticeably (large ratio); pull a block of cork and it barely changes width (near zero).
Precisely, for a bar under simple tension, Poisson's ratio is nu = -(transverse strain)/(axial strain), the minus sign making nu positive for the usual case where a stretch causes a sideways contraction. For an isotropic elastic material it relates to the Lamé parameters as nu = lambda / [2(lambda + mu)], and together with Young's modulus E it fully specifies the isotropic elastic response. Thermodynamic stability restricts it to the range -1 < nu < 1/2 for an isotropic solid. The upper limit nu = 1/2 is the incompressible case -- stretch causes exactly enough thinning to keep the volume fixed.
Most metals have nu near 0.3, rubber approaches 0.5 (nearly incompressible), and cork is close to 0 (which is why a cork pushes into a bottle without jamming sideways). A few engineered auxetic materials, and some foams, have negative Poisson's ratio -- they get fatter when stretched, expanding sideways. Poisson's ratio is one of the two everyday elastic constants (with Young's modulus) that engineers quote for any structural material.
Steel has nu about 0.30: stretch a steel wire by 1 percent along its length and it thins by about 0.30 percent across its diameter. Rubber, with nu near 0.5, thins by almost the full 0.5 percent -- because it must conserve its volume.
Poisson's ratio is minus the transverse strain over the axial strain; 0.5 means incompressible.
A value of exactly 1/2 does not mean 'infinitely stiff' -- it means incompressible, i.e. the bulk modulus diverges while the shear modulus stays finite. Rubber is nearly incompressible yet very easy to shear, which is why it is soft to the touch but hard to squash smaller.