# Poisson's Ratio

## Definition

When a material is elongated or compressed along one axis, the Poisson's ratio, $\nu$, of a material is the ratio of the Strain in the directions perpendicular to the axis, divided by the Strain along that axis. Essentially, this dimensionless number describes the tendency of a contraction or expansion in one dimension to cause contraction or expansion in other dimensions. Most materials have a positive Poisson's ratio, meaning other dimensions will contract in response to elongation along one dimension, and vice versa. Most materials have a Poisson's ratio near 0.3, with more rubbery materials approaching 0.5. [3] Materials with negative Poisson's ratio were first reported in 1987 by Lakes. Example materials of this type include some foams, honeycomb structures produced by Prall and Lakes,[3] and patterned arrays of circular and elliptical holes in an elastomer (Bertoldi).[3] In these last two cases, local buckling of the structure walls leading to volume reduction is directly observed to cause a negative Poisson's ratio. Figure 1 shows a schematic of local buckling in Bertoldi's material.

Important Limits

For an isotropic, unconstrained three dimensional elastic material, Poisson's ratio must range from -1 to 1/2.[3]

## References

[1]http://silver.neep.wisc.edu/~lakes/PoissonIntro.html


[2] '"Negative Poisson’s Ratio Behavior Induced by an Elastic Instability". Katia Bertoldi, Pedro M. Reis, Stephen Willshaw, and Tom Mullin. Adv. Mater. 2009, 21, 1–6.

[3] "Properties of a chiral honeycomb with a Poisson's ratio -1" D. Prall, R. S. Lakes. Int. J. of Mechanical Sciences, 39, 305-314, (1996)