Poisson's Ratio Explained with Examples

Poisson’s ratio is one of the elastic constants that describe how a solid deforms under load. Together with Young’s modulus it fixes the shear modulus and the bulk modulus of an isotropic material, so it appears in shaft design, pressure vessel calculations, strain gauge work and every finite element model. This guide explains the definition, typical values, the links to other elastic constants and how the ratio is measured, then works through a numerical example.

What is Poisson’s ratio?

When a prismatic bar is pulled along its length, it stretches in that direction. At the same time its width and thickness shrink. The French mathematician Siméon Denis Poisson studied this lateral contraction, and the ratio is named after him.

Two strains are involved:

• Axial (longitudinal) strain: ε_axial = ΔL / L, the change in length divided by the original length, measured along the load.
• Lateral (transverse) strain: ε_lateral = Δd / d, the change in a transverse dimension such as diameter or width, divided by its original value.

In tension, ε_axial is positive and ε_lateral is negative. In compression the signs reverse. The definition adds a minus sign so that the ratio is positive for ordinary materials:

$$\nu = -\frac{\varepsilon_{lateral}}{\varepsilon_{axial}}$$

Poisson’s ratio is dimensionless. It is constant only in the linear elastic range. Once the material yields, plastic flow takes place at almost constant volume and the apparent ratio rises toward 0.5.

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Typical values of Poisson’s ratio

The ratio depends mainly on the type of bonding and structure of the material. Values below are typical figures from mechanics and materials textbooks; use supplier data or the material standard for design.

Rubber is nearly incompressible, so its volume hardly changes and ν approaches 0.5. Cork has a cellular structure that collapses without bulging sideways, which is why a cork can be pushed into a bottle neck easily. Some engineered foams and lattice structures, called auxetic materials, have a negative Poisson’s ratio: they get wider when stretched.

Poisson’s ratio and the other elastic constants

For a homogeneous, isotropic, linear elastic material only two elastic constants are independent. If Young’s modulus E and Poisson’s ratio ν are known, the shear modulus G and bulk modulus K follow:

$$G = \frac{E}{2(1+\nu)} \qquad K = \frac{E}{3(1-2\nu)}$$

These relations explain the allowed range. For G to be positive, ν must be greater than −1. For K to be positive, ν must be less than 0.5. For steel with E = 200 GPa and ν = 0.3, G is about 77 GPa and K is about 167 GPa.

Volume change under axial load

A bar under uniaxial stress changes volume by a fraction:

$$\frac{\Delta V}{V} = \varepsilon_{axial}\,(1 - 2\nu)$$

With ν = 0.5 the volume change is zero. With ν = 0.3 a stretched steel bar gains volume equal to 40 percent of its axial strain.

Multiaxial loading

In a body loaded in several directions, Poisson’s ratio couples the strains. The generalized Hooke’s law for normal strain in the x direction is:

$$\varepsilon_x = \frac{1}{E}\left[\sigma_x - \nu\left(\sigma_y + \sigma_z\right)\right]$$

This is why a thin plate restrained from contracting sideways is stiffer than a free bar, and why the hoop and axial strains in a pressurized cylinder each depend on both stresses. Introductory treatment of the stresses themselves is in the guide to stress and strain.

How Poisson’s ratio is measured

ASTM E132 describes a room temperature tension test for structural materials. A flat specimen is loaded in the elastic range while axial and transverse strains are measured, normally with bonded strain gauges or extensometers on both axes. The ratio is taken from the slopes of the transverse strain and axial strain plotted against stress.

Points to watch during measurement:

• Keep the load in the linear elastic range; yielding changes the ratio.
• Align the specimen carefully, because bending adds strain on one face and distorts readings.
• Use gauges with matched gauge factors and correct for the transverse sensitivity of the gauge where required.

Dynamic methods, which compute ν from measured longitudinal and shear wave speeds or resonant frequencies, are also used, especially for ceramics and concrete.

The diameter change is tiny compared with the elongation, which is why it is rarely noticed in practice. It still matters for press fits, seals and bolts, where a few micrometres change the contact pressure.

Where Poisson’s ratio is used

• Pressure vessels and pipes: hoop and axial strains of a cylinder depend on ν, which affects the diameter growth of a pipe under pressure. See the basics of pressure vessels.
• Strain gauges and load cells: gauges are often arranged with some elements along the load and some across it. The transverse gauges read the Poisson strain and increase bridge output. The pressure measurement articles on InstrumentationBlog.in show how such gauges are used in sensors.
• Finite element analysis: every linear static model needs E and ν for each material. An incorrect ν, or a value too close to 0.5 with ordinary elements, can make results inaccurate. The finite element analysis basics guide covers material input.
• Geotechnical and concrete design: ν of soil and concrete enters settlement and slab calculations.
• Rubber mounts and seals: near incompressible behaviour means a rubber block loaded in compression bulges sideways, so its stiffness depends strongly on its shape.

Common mistakes

• Dropping the minus sign and quoting a negative ratio for steel.
• Applying the isotropic relations G = E / (2(1 + ν)) to wood, composites or rolled sheet, which are anisotropic and have different ratios in different directions.
• Using ν = 0.5 exactly in a finite element model with standard elements, which can cause numerical locking.
• Confusing Poisson’s ratio with the Poisson distribution in statistics, which is unrelated.

Related reading

• Stress and Strain: Definitions, Types and Hooke’s Law
• Factor of Safety in Machine Design Explained
• Finite Element Analysis (FEA): Basic Steps Explained