# 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.

MaterialPoisson'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
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

## Related Topics

• Miscellaneous - Engineering related topics like Beaufort Wind Scale, CE-marking, drawing standards and more
• Mechanics - Forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more

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