What Is Poisson’s Ratio?

Poisson’s Ratio is defined as the ratio between the lateral contraction (expansion) strain to the longitudinal extension (contraction) strain when an object is subjected to an external tensile (compressive) force along its longitudinal axis.

When a solid bar is subjected to tensile force along its longitudinal axis, the length of the bar increases in the direction of application of the force. Since the molecules in solids are closely pa mecked, the density of the solids remains practically constant. So, to keep its volume unchanged, the solid contracts in its lateral direction. The ratio of changes in dimensions to its original dimensions in the longitudinal and lateral directions are called the ‘longitudinal strain’ and the ‘lateral strain,’ respectively. Poisson’s Ratio is expressed as the ratio of this lateral strain to the longitudinal strain, with a negative sign before it.

Poisson’s Ratio is named after the French mathematician Simeon Denis Poisson. Poisson’s Ratio plays a vital role in engineering design. It is used to calculate the stress and deflection of beams, shafts, rotors, etc. when subjected to external loads and thus helps to optimize engineering designs.

The technical definition of Poisson’s Ratio

In technical terms, Poisson’s ratio is a ratio of the transverse contraction or expansion strain (perpendicular) against the longitudinal strain (i.e. in the same direction as the applied force).

Poisson’s ratio is denoted by the Greek letter nu (μ).

Mathematically:

μ = – ε_t / ε_l

Given that:

μ = Poisson’s ratio

ε_t = transverse (perpendicular) strain

ε_l = longitudinal strain

The minus sign allows for the fact that stretching deformation is taken as positive and compressive deformation is negative.

The longitudinal strain can be calculated by using the following formula:

ε_l = dl / L                             

Given that:

ε_l = longitudinal strain

dl = difference in length (m)

L = static length (m)

The transverse strain can be calculated by using the following formula:

ε_t = dr / r                             

Given that:

ε_t = transverse strain

dr = difference in radius (m)

r = initial radius (m)

Poisson’s ratio is the ratio of transverse contraction strain to longitudinal extension strain in the direction of stretching force. Tensile deformation is considered positive and compressive deformation is considered negative. The definition of Poisson’s ratio contains a minus sign so that normal materials have a positive ratio. Poisson’s ratio, also called Poisson ratio or the Poisson coefficient, or coefficient de Poisson, is usually represented as a lower case Greek nu, n.

Common values of Poisson’s Ratio

This table contains values for some commonly used materials: