# Poisson's Ratio

When a sample of material is stretched in one direction it tends to get thinner in the lateral direction - and if a sample is compressed in one direction it tends to get thicker in the lateral direction.

** Poisson's ratio ** is

- the ratio of the relative contraction strain (transverse, lateral or radial strain) normal to the applied load - to the relative extension strain (or axial strain) in the direction of the applied load

Poisson's Ratio can be expressed as

μ = - ε_{ t }/ ε_{ l }(1)

where

μ = Poisson's ratio

ε_{ t }= transverse strain (m/m, ft/ft)

ε_{ l }= longitudinal or axial strain (m/m, ft/ft)

Strain is defined as "deformation of a solid due to stress".

** Longitudinal ** (or axial) strain can be expressed as

ε_{ l }= dl / L (2)

where

ε_{ l }= longitudinal or axial strain (dimensionless - or m/m, ft/ft)

dl = change in length (m, ft)

L = initial length (m, ft)

** Contraction ** (or transverse, lateral or radial) strain can be expressed as

* ε _{ t } = dr / r (3) *

* where *

* ε _{ t } = transverse, lateral or radial strain (dimensionless - or m/m, ft/ft) *

* dr = change in radius (m, ft) *

* r = initial radius (m, ft) *

Eq. 1, 2 and 3 can be combined to

*μ = - ( dr / r ) / ( dl / L ) (4) *

### Example - Stretching Aluminum

An aluminum bar with length * 10 m * and radius * 100 mm (100 10 ^{-3} m) * is stretched

*5 mm (5 10*. The radial contraction in lateral direction can be rearranged to

^{-3}m)* dr = - μ r dl / L (5) *

With Poisson's ratio for aluminum * 0.334 * - the contraction can be calculated as

* dr = - 0.334 ( 100 10 ^{-3} m ) ( 5 10^{-3} m) / (10 m) *

* = 1.7 10 ^{ -5 } m *

* = 0.017 mm *

### Poisson's Ratios for Common Materials

For most common materials the Poisson's ratio is in the range * 0 - 0.5 * . Typical Poisson's Ratios for some common materials are indicated below.

Material | Poisson's Ratio - μ - |
---|---|

Upper limit |
0.5 |

Aluminum | 0.334 |

Aluminum, 6061-T6 | 0.35 |

Aluminum, 2024-T4 | 0.32 |

Beryllium Copper | 0.285 |

Brass, 70-30 | 0.331 |

Brass, cast | 0.357 |

Bronze | 0.34 |

Clay | 0.41 |

Concrete | 0.1 - 0.2 |

Copper | 0.355 |

Cork | 0 |

Glass, Soda | 0.22 |

Glass, Float | 0.2 - 0.27 |

Granite | 0.2 - 0.3 |

Ice | 0.33 |

Inconel | 0.27 - 0.38 |

Iron, Cast - gray | 0.211 |

Iron, Cast | 0.22 - 0.30 |

Iron, Ductile | 0.26 - 0.31 |

Iron, Malleable | 0.271 |

Lead | 0.431 |

Limestone | 0.2 - 0.3 |

Magnesium | 0.35 |

Magnesium Alloy | 0.281 |

Marble | 0.2 - 0.3 |

Molybdenum | 0.307 |

Monel metal | 0.315 |

Nickel Silver | 0.322 |

Nickel Steel | 0.291 |

Polystyrene | 0.34 |

Phosphor Bronze | 0.359 |

Rubber | 0.48 - ~0.5 |

Sand | 0.29 |

Sandy loam | 0.31 |

Sandy clay | 0.37 |

Stainless Steel 18-8 | 0.305 |

Steel, cast | 0.265 |

Steel, Cold-rolled | 0.287 |

Steel, high carbon | 0.295 |

Steel, mild | 0.303 |

Titanium (99.0 Ti) | 0.32 |

Wrought iron | 0.278 |

Z-nickel | 0.36 |

Zinc | 0.331 |

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