Poisson's Ratio – Definition, Values for Materials, and Applications

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.