# Poisson's Ratio

on . Posted in Classical Mechanics

Poisson's ratio, abbreviated as Po, $$\mu$$ or $$\nu$$, a dimensionless number, relates the lateral and axial strains of an elastic material when it is subjected to an external stress.

Poisson's ratio is a measure of the degree of deformation that a material undergoes when subjected to an external stress.  It is always negative or between 0 and 0.5 for most materials, indicating that when a material is compressed or stretched, it will contract or expand laterally.  The value of Poisson's ratio depends on the type of material and its structure, and it is an important parameter in engineering and materials science, especially in the design of structures that require elasticity and resilience, such as buildings, bridges, and aircraft.

### Poisson's ratio Interpretation

Poisson's ratio ranges between -1 and 0.5 for typical materials.

• If ν = −1  -  It indicates that the material experiences maximum contraction in the transverse direction when compressed longitudinally.
• If ν = 0.5t means that the material does not change in lateral size at all when compressed or stretched longitudinally (this is rare and only occurs in a few materials like incompressible rubber).
• If ν = 0 It suggests that the material undergoes equal contraction and expansion in the transverse and longitudinal directions.
• If ν is close to 0.5  -  The material is considered incompressible in the plane of deformation.

Different materials have different Poisson's ratios.  For most common materials, Poisson's ratio falls in the range of 0 to 0.5.  Metals generally have Poisson's ratios around 0.3, while rubber-like materials can have Poisson's ratios close to 0.5.

### Poisson's Ratio formula

$$\mu \;=\; \varepsilon_t \;/\; \varepsilon_a$$     (Poisson's Ratio)

$$\varepsilon_t \;=\; \mu \; \varepsilon_a$$

$$\varepsilon_a \;=\; \varepsilon_t \;/\; \mu$$

Symbol English Metric
$$\mu$$  (Greek symbol mu) = Poisson's Ratio $$dimensionless$$
$$\epsilon_t$$  (Greek symbol epsilon) = lateral or transverse strain (direction of load) $$in\;/\;in$$ $$mm\;/\;mm$$
$$\epsilon_a$$  (Greek symbol epsilon) = axial or longitudinal strain (right angle to load) $$in\;/\;in$$ $$mm\;/\;mm$$