# Poisson's ratio

## When a material is stretched in one direction it tends to get thinner in the other two directions

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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

εlongitudinal or axial strain (dimensionless - or m/m, ft/ft)_{l}=

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 (2)*

*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)*

### 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 calculated by combining eq.

^{-3}m)*(1)*and

*(2)*to

*μ = - ( dr / r) / (dl / L) (3)*

*-* and rearranging to

*dr = - μ r dl / L (3b)*

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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