# Poisson Ratio – its 3 [Formulas, Ratios Of Negative & Ratio in Common Materials]

## Poisson Ratio

Poisson ratio is the negative of the ratio of the lateral strain to the axial strain or measure of the Poisson effect, which describes the expansion of material in directions perpendicular to the direction of loading and in direction of stretching forces and is transverse contraction strain to longitudinal extension strain.

The material will elongate on the axis of the load if the tensile load is applied to the material and if the load is compressive the axial dimension will decrease.

A corresponding lateral contraction must occur if the volume is constant and to the axial strain, this lateral change will bear a fixed relationship and the relationship or ratio of lateral to axial strain is called this ratio.

In engineering analysis for determining the stress and deflection properties of materials, this ratio is required constant and it is also constant for structures like beams, rotating discs, plates, and shells.

The magnitude of stresses and strains and direction of loading all have their effects on the Poisson ratio with plastic when the temperature changes. The design of structures like 2D & 3D the application of the Poisson ratio is frequently required.

For more than 200 years Poisson ratio has been a basic principle of engineering and allows engineers to identify how much a material can be compressed and stretched and before it collapses how much pressure it will withstand.

### Formula For Poisson Ratio:

The equation for the Poisson ratio is;

μ = – εt / εl

where,

μ is the Poisson ratio

εt is the transverse strain in m/m, ft/ft

εl is the longitudinal or axial strain in m/m, ft/ft

As,

#### Longitudinal or axial strain can be expressed as;

εl = dl / L

εl is the longitudinal or axial strain in m/m, ft/ft or dimensionless

dl is the change in length along the direction of force in m, ft

L is the initial length along the direction of force in m, ft

#### Transverse or lateral strain can be expressed as;

εt = dr / r

εt is the transverse or lateral strain in m/m, ft/ft or dimensionless

dr is the change in radius in m, ft

r is the initial radius in m, ft

### Poisson Ratios For Common Materials;

There are following common materials with their Poisson ratios given in the table below;

 Materials Poisson’s Ratio– μ – Upper limit 0.5 Aluminum 0.334 Rubber 0.48 – 0.5 Indium 0.45 Gold 0.42 Lead 0.44 Copper 0.37 Polystyrene 0.34 Brass 0.33 Ice 0.33 Polystyrene foam 0.3 Stainless steel 0.30 Steel 0.29 Tungsten 0.28 Zinc 0.25 Boron 0.08 Beryllium 0.03 Re-entrant foam -0.7 Isotropic lower limit -1

### Negative Poisson Ratio Materials:

Some auxetic materials show a negative Poisson ratio and the transverse strain in the material will actually be positive when subjected to positive strain in a longitudinal axis and this is due to uniquely orient and hinged molecular bonds.

During a compression creep test certain wood solid wood types display negative Poisson ratio and compression creep test shows positive Poisson ratios.

During constant loading Poisson ratio for wood is time-dependent and this means that strain in the axial and transverse direction don’t increase in the same rate and materials with a negative Poisson ratio have been called rubber, auxetic materials, and dilational materials. Some of auxetic materials are;

1. Tendons with a limited range of motions.
2. Polymers such as Gore-tex.
3. Polyurethane and graphene.

Negative Poisson ratio materials give the benefit of high energy absorption and possess high resistance and used for packing materials, medical knee pads and footwears, etc.

### Applications Of Poisson Effects:

Poisson effects have a considerable influence is in pressurized pipe flow and the air or liquid inside the pipe is highly pressurized and on the inside of pipe it exerts a uniform force and resulting in a hoop stress.

This will cause the pipe to increase in diameter and length is decreased and a decrease in length can have a noticeable effect upon the pipe joints.

It might be helpful in improving the fracture of the composite and may be used in body armors.

### Points To Remember:

There are following points to be remembered such as;

1. The lateral strain will be compressive if the longitudinal strain is tensile.
2. The longitudinal strain will be compressive if the lateral strain is tensile.
3. A longitudinal strain in the direction of the applied load is every time accompanied by an equal and opposite lateral strain.