Poisson's Ratio
When a material is stretched in one direction it tends to get thinner in the other two directions.
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 = - εt / εl (1)
where
μ = Poisson's ratio
εt = transverse strain (m/m, ft/ft)
εl = 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
εl = dl / L (2)
where
εl = 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
εt = dr / r (2)
where
εt = 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 10-3 m). The radial contraction in lateral direction can be calculated by combining eq. (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 |