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# 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 = - 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 10-3 m) . The radial contraction in lateral direction can be rearranged to

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.

Poisson's Ratios common Materials
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

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• ### Metals and Alloys - Young's Modulus of Elasticity

Elastic properties and Young's modulus for metals and alloys like cast iron, carbon steel and more.
• ### Modulus of Rigidity

Shear Modulus (Modulus of Rigidity) is the elasticity coefficient for shearing or torsion force.
• ### Poisson's Ratios Metals

Some metals and their Poisson's Ratios.
• ### Ratios and Proportions

The relative values between quantities - ratios and proportions.
• ### Stress, Strain and Young's Modulus

Stress is force per unit area - strain is the deformation of a solid due to stress.
• ### Young's Modulus, Tensile Strength and Yield Strength Values for some Materials

Young's Modulus (or Tensile Modulus alt. Modulus of Elasticity) and Ultimate Tensile Strength and Yield Strength for materials like steel, glass, wood and many more.

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## Citation

• The Engineering ToolBox (2008). Poisson's Ratio. [online] Available at: https://www.engineeringtoolbox.com/poissons-ratio-d_1224.html [Accessed Day Month Year].

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7.25.9

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