Poisson’s Ratio

Poisson’s ratio is the negative of the ratio of transverse strain to lateral or axial strain in physics. It is named for Siméon Poisson and is represented by the Greek letter ‘nu.’ For values of these changes, it is the ratio of transversal expansion to the amount of axial compression. The Poisson effect occurs when a material expands in directions perpendicular to the compression direction. Poisson’s effect is used in various fields, including structural geology, pressurised pipe flow, bottle corks, and automobile mechanics.

What is Poisson’s ratio?

In the direction of the stretching force, it is the ratio of transverse contraction strain to longitudinal extension strain as described by Poisson. Here, compressive deformation is considered negative, whereas tensile deformation is positive. The definition of Poisson’s ratio includes a negative sign, indicating that standard materials have a positive ratio. The Poisson ratio, also known as the Poisson coefficient, is usually represented as nu, n in lowercase.

Poisson’s ratio formula

Poisson’s ratio is represented numerically below by using its definition:

The Poisson’s Ratio is calculated as

μ = – εt / εl                   (1)          

where

μ = Poisson’s ratio

εt = transverse strain 

εl = axial or longitudinal strain

“Stress-induced deformation of a solid” is how strain is defined.

Longitudinal (or axial) strain can be stated as

εl = dl / L                          (2)    

where,

εl =longitudinal or axial strain 

dl = change in length 

L = initial length

The strain of contraction (also known as transverse, lateral, or radial strain) is stated as

εt = dr / r                         (2)     

where,

εt = transverse, lateral or radial strain

dr = change in radius 

r = initial radius 

Example – Stretching Aluminium

A 10 m long metal bar of aluminium with a radius of 100 mm (100×10-3 m) is stretched 5 mm (5×10-3 m). Combining equations (1) and (2), the radial contraction in the lateral direction may be calculated.

μ = – (dr / r) / (dl/L)                     

– and rearranging to

dr = – μ r dl / L (3b)

The contraction may be computed using Poisson’s ratio of 0.334 for aluminium.

dr = – 0.334 (5×10-3 m) (100×10-3 m) / (10 m)

    = 0.017 mm      

Poisson’s Ratio Unit

The ratio of two strains is known as Poisson’s ratio. The strain is dimensionless in both the longitudinal and lateral directions. As a result, Poisson’s ratio has no dimensions. It does not have a unit.

Application of Poison effect

• In the case of pressurised pipe flow, Poisson’s effect has a significant impact. When air or liquid inside a pipe is under high pressure, it exerts a consistent force on the inside of the tube, causing hoop stress in the pipe material. This hoop stress will force the pipe to expand in diameter while shrinking in length due to Poisson’s effect. Because the effect accumulates for each piece of pipe joined in series, the decrease in length might substantially affect the pipe joints. A restrained joint might be wrenched apart or fail in other ways.

• In the field of structural geology, Poisson’s effect has another use. When stressed, rocks, like most materials, are subject to Poisson’s effect. Over geological timescales, excessive erosion or sedimentation of the Earth’s crust can create or remove significant vertical strains on the underlying rock. As a direct result of the imposed stress, this rock will expand or contract vertically and deform horizontally due to Poisson’s effect. Joints and dormant tensions in the stone might be affected or formed by this change in horizontal strain.

• Most vehicle mechanics know that extracting a rubber hose (such as a coolant hose) from a metal pipe stub is difficult due to the stress causing the hose’s diameter to shrink, firmly grasping the stub. Stockings may be pushed off stubs more efficiently with a broad flat blade than a wide flat blade.

Conclusion

In material science and engineering mechanics, the Poisson’s ratio of a material is a critical metric. When a force is applied to a bar, it deforms in the axial (longitudinal) direction (elongates or compresses). At the same time, there is a deformation in the transverse (width) direction. Poisson’s ratio relates these variations in the transverse and axial directions. The effect is known as Poisson’s effect, and it is named after Simeon Poisson, a French mathematician and scientist. Most materials, Poisson’s ratio lies between 0.0 and 0.5. Poisson’s ratio is near 0.5 for soft materials like rubber, where the bulk modulus is substantially more significant than the shear modulus. Poisson’s ratio is 0 in open-cell polymer foams because the cells collapse with compression. Several popular solids have Poisson’s ratios of 0.2-0.3.