The Theory on Poisson’s Ratio

Poisson’s ratio display style nu (𝛎) is a measure of the Poisson effect, which is the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading, in material science and solid mechanics. 

Poisson’s ratio is the inverse of the transverse strain to axial strain ratio. Displaystyle nu (𝛎) is the amount of transversal elongation divided by the amount of axial compression for small values of these changes. 

The Poisson’s ratio of most materials ranges between 0.0 and 0.5. Poisson’s ratio is close to 0.5 for soft materials such as rubber, where the bulk modulus is much higher than the shear modulus. 

Poisson’s ratio is near zero for open-cell polymer foams because the cells tend to collapse in compression. Many common solids have Poisson’s ratios in the 0.2-0.3 range. 

Siméon Poisson, a French mathematician and physicist, inspired the ratio’s name.

What is Poisson’s Ratio?

Poisson’s ratio is a material constant that describes the relationship between stress in one direction and elastic strain in another. Stress is defined as an external force applied to a material, whereas strain is defined as the change in shape caused by that force.

Vertical strain is calculated as the proportional difference in L (length), and horizontal strain is calculated in the same way using the radial difference (d). 

The original length and diameter are denoted by L and d, respectively, and the new measurements are denoted by L’ and d’.

Poisson’s understanding of this ratio was based on experiments with brass rods at the time. The radius of the rod was observed to decrease as it was pulled in an axial direction. 

To estimate the strain that is happening to the wire, we can compute unit-less values that measure the strain in the axial and radial, or outwards from the centre, directions.

The negative ratio of these two values is known as Poisson’s ratio. 

He recognised that applying stress in one direction resulted in an opposite response in the opposite direction for most materials. This is why he multiplied his ratio by -1 to ensure a positive final value.

Why is the Poisson’s Ratio is positive?

When we stretch common materials, their cross-sections become narrower. This phenomenon can also be seen when stretching a piece of rubber.

Furthermore, in the continuum view, most materials resist a change in volume as determined by bulk absolute value K. (also called B at some places).

Furthermore, as determined by shear absolute value G, they resist a change in shape. If we look at it structurally, this is due to the usual positive Poisson’s Ratio of interatomic bonds that realign with deformation.

Poisson’s Ratio in waves and deformation

A material’s Poisson’s ratio influences the speed of propagation and reflection of stress waves. The ratio of compressional to shear wave speed is important in geological applications for determining the nature of rock deep within the Earth. This wave speed ratio is determined by the Poisson’s ratio. Poisson’s ratio also influences the decay of stress with distance according to Saint Venant’s principle, as well as the stress distribution around holes and cracks

Poisson’s Ratio in viscoelastic materials

In the context of transient tests such as creep and stress relaxation, the Poisson’s ratio in a viscoelastic material is time dependent. Poisson’s ratio may depend on frequency and have an associated phase angle if the deformation is sinusoidal in time. In a viscoelastic solid, the transverse strain may be out of phase with the longitudinal strain. Poisson’s ratio is on the order of 1/3 for polymers in the glassy regime. The stiffness decreases dramatically at higher temperatures (or longer times or lower frequency) sufficient for rubbery behaviour, and the Poisson’s ratio approaches ½.

Poisson’s Ratio and phase transformation

In the vicinity of a phase transformation, Poisson’s ratio can change dramatically. The bulk modulus typically softens near a phase transition, but the shear modulus does not change significantly. The Poisson’s ratio then decreases and can reach negative values in the vicinity of a phase transformation.

Conclusion

Poisson’s ratio display style nu ν is a measure of the Poisson effect, which is the deformation of a material in directions perpendicular to the specific direction of loading, in material science and solid mechanics. Poisson’s ratio is the inverse of the transverse strain to axial strain ratio. The negative ratio of these two values is known as Poisson’s ratio. A material’s Poisson’s ratio influences the speed of propagation and reflection of stress waves. This wave speed ratio is determined by the Poisson’s ratio. Poisson’s Ratio and phase transformation In the vicinity of a phase transformation, Poisson’s ratio can change dramatically. The Poisson’s ratio then decreases and can reach negative values in the vicinity of a phase transformation.