# Poisson's Ratio

## Definition

When a material is elongated or compressed along one axis, the Poisson's ratio, <math>\nu</math>, 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. Figure 1 demonstrates this concept.

Most materials have a Poisson's ratio near 0.3, with more rubbery materials approaching 0.5. [3] Auxetic materials, or materials with a negative Poisson's ratio were first reported in 1987 by Lakes.[2] 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 (i.e., repeatable collapse) 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. The material compresses uniformly until a critical applied strain is reached. At this strain, an elastic instability develops, and the structure experiences uniform, repeatable local buckling of the walls. Figures 3 and 4 show Prall and Lakes' honeycomb structures. Figure 3 shows a "regular" honeycomb shape, with a positive Poisson's ratio, and a "reentrant" honeycomb shape, with a negative Poisson's ratio. Figure 4 shows the same groups chiral honey comb structures in a) a schematic and b) a photograph of the fabricated structure. [3]

*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)

## Keyword in References

Homogeneous flow of metallic glasses: A free volume perspective