Engineers often need to observe how different objects respond to forces or pressures within real-world situations. One such observation is how the length of an object expands or contracts under the application of a force.

This physical phenomenon is known as strain and is defined as the change in length divided by the total length. **Poisson's ratio** quantifies the change in length along two orthogonal directions during the application of a force. This quantity can be calculated using a simple formula.

## Poisson Ratio Formula

**Poisson's ratio** is the ratio of the relative contraction strain (that is, the transverse, lateral or radial strain) *perpendicular to* the applied load to the relative extension strain (that is, the axial strain) *in the direction of* the applied load. Poisson's ratio can be expressed as

**μ = –ε _{t} / ε_{l}.**

where μ = Poisson's ratio, ε_{t} = transverse strain (m/m, or ft/ft) and ε_{l} = longitudinal or axial strain (again m/m or ft/ft).

Young's modulus and Poisson's ratio are among the most important quantities in the area of stress and strain engineering.

## Poisson's Ratio Strength of Materials

## Longitudinal Strain

## Transverse Strain

## Deriving the Formula

Think about how a force exerts strain along two orthogonal directions of an object. When a force is applied to an object, it gets shorter along the direction of the force (longitudinal) but gets longer along the orthogonal (transverse) direction. For example, when a car drives over a bridge, it applies a force to the bridge's vertical supporting steel beams. This means that the beams get a bit shorter as they are compressed in the vertical direction but get a bit thicker in the horizontal direction.

Calculate the longitudinal strain, ε_{l}, using the formula ε_{l} = - dL/L , where dL is the change in length along the direction of force, and L is the original length along the direction of the force. Following the bridge example, if a steel beam supporting the bridge is approximately 100 meters tall, and the change in length is 0.01 meters, then the longitudinal strain is ε_{l} = –0.01/100 = –0.0001.

Because strain is a length divided by a length, the quantity is dimensionless and has no units. Note that a minus sign is used in this length change, as the beam is getting shorter by 0.01 meters.

Calculate the transverse strain, ε_{t}, using the formula **ε _{t}** = dLt/Lt, where dLt is the change in length along the direction orthogonal to the force, and Lt is the original length orthogonal to the force. Following the bridge example, if the steel beam expands by approximately 0.0000025 meters in the transverse direction and its original width was 0.1 meters, then the transverse strain is ε

_{t}= 0.0000025/0.1 = 0.000025.

Write down the formula for Poisson's ratio: **μ = –ε _{t} / ε_{l}.** Again, note that Poisson's ratio is dividing two dimensionless quantities, and therefore the result is dimensionless and has no units. Continuing with the example of a car going over a bridge and the effect on the supporting steel beams, the Poisson's ratio in this case is

**μ**= –(0.000025/–0.0001) = 0.25.

This is close to the tabulated value of 0.265 for cast steel.

## Poisson's Ratio for Common Materials

Most everyday building materials have a μ in the range of 0 to 0.50. Rubber is close to the high end; lead and clay are both over 0.40. Steel tends to be closer to 0.30 and iron derivatives lower still, in the 0.20 to 0.30 range. The lower the number, the less amenable to "stretching" forces the material in question tends to be.

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