Poisson's ratio

Poisson's ratio, denoted by the Greek letter ν {displaystyle u } ('nu'), and named after the French mathematician and physicist Siméon Poisson, is the negative of the ratio of (signed) transverse strain to (signed) axial strain. For small values of these changes, ν {displaystyle u } is the amount of transversal expansion divided by the amount of axial compression.There are two valid solutions. The plus sign leads to ν ≥ 0 {displaystyle u geq 0} . Poisson's ratio, denoted by the Greek letter ν {displaystyle u } ('nu'), and named after the French mathematician and physicist Siméon Poisson, is the negative of the ratio of (signed) transverse strain to (signed) axial strain. For small values of these changes, ν {displaystyle u } is the amount of transversal expansion divided by the amount of axial compression. 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, 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, isotropic, linear 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. 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) exhibit values of about 0.3, increasing to 0.5 for post-yield deformation which occurs largely at constant volume. Rubber has a Poisson ratio of nearly 0.5. Cork's Poisson ratio is close to 0, showing very little lateral expansion when compressed. Some materials, e.g. some polymer foams, origami folds, and certain cells can exhibit negative Poisson's ratio, and are referred to as auxetic materials. If these auxetic materials are stretched in one direction, they become thicker in the perpendicular direction. In contrast, some anisotropic materials, such as carbon nanotubes, zigzag-based folded sheet materials, and honeycomb auxetic metamaterials 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 along the axial direction (the x axis in the diagram below):