Engineering · Materials mechanics

Poisson’s ratio

In a uniaxial test, Poisson’s ratio is the negative ratio of transverse strain to axial strain.

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Formula

ν = - \frac{ε _{quer}}{ε _{axial}}

ν dimensionless; both strains dimensionless

Variables & units

Variables & units
Symbol	Meaning	Unit
$ν$	Poisson’s ratio in the stated material range	dimensionless
$εquer$	Strain perpendicular to loading	dimensionless
$εaxial$	Strain along the loading direction	dimensionless

LaTeX

\nu=-\frac{\varepsilon_\mathrm{quer}}{\varepsilon_\mathrm{axial}}

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Derivation & background

Ordinary isotropic materials typically contract laterally in tension, which motivates the minus sign. The definition describes measured material response in the small-strain linear-elastic range. A single scalar is insufficient for anisotropic materials, and the value should not be treated as universal across all conditions.

Exam blueprint

Validity range

Derivation steps

In a uniaxial test, Poisson’s ratio is the negative ratio of transverse strain to axial strain.

1State the model assumptions and list quantities with units.
2Substitute into ν = −εquer/εaxial; convert quantities to SI first.
3For εaxial=0.002 and εtransverse=−0.0006, ν=−(−0.0006)/0.002=0.30.

Rearrangements

Base equation for the target quantity

\nu=-\frac{\varepsilon_\mathrm{quer}}{\varepsilon_\mathrm{axial}}

Preserve signs, boundary conditions and units when rearranging.

Task variant

Work through the numerical example for “Poisson’s ratio” and check its unit and magnitude.

For εaxial=0.002 and εtransverse=−0.0006, ν=−(−0.0006)/0.002=0.30.

Common mistakes

Substituting a related quantity that shares the same symbol.

Use variable definitions and check: ν dimensionless; both strains dimensionless.

Exam context

Estimate lateral contraction and parameterize elastic material models.

Formula cluster

Materials mechanics

In a uniaxial test, Poisson’s ratio is the negative ratio of transverse strain to axial strain.

E = σ/ε = FL₀/(A₀ΔL)

Young's modulus

σ_m = F/A

Average axial normal stress

q⃗ = −k∇T; Q̇ = −kA·dT/dx

Fourier’s law of heat conduction

Worked example

For εaxial=0.002 and εtransverse=−0.0006, ν=−(−0.0006)/0.002=0.30.

Applications

Estimate lateral contraction and parameterize elastic material models.

How do you calculate with Poisson’s ratio?

Here is how to work through a typical Poisson’s ratio (ν = −εquer/εaxial) task step by step:

1
Task
Work through the numerical example for “Poisson’s ratio” and check its unit and magnitude.
Solution path
For εaxial=0.002 and εtransverse=−0.0006, ν=−(−0.0006)/0.002=0.30.

Frequently asked questions about Poisson’s ratio

What does the minus sign mean?+

In an ordinary tensile test, axial strain is positive while transverse strain is usually negative because the specimen narrows. The minus sign therefore makes Poisson’s ratio typically positive. Under compression both strain signs reverse, so their negative ratio remains positive in an ordinary case. Poisson’s ratio is a dimensionless measured material response in the small-strain elastic range. Identify axial and transverse directions before applying the minus sign. Anisotropic materials need directional elastic constants; one scalar ratio cannot describe every orientation or nonlinear response.

How do I determine ν from measurements?+

Measure relative length change along the load and perpendicular to it. Divide transverse strain by axial strain and apply the minus sign. For axial 0.002 and transverse −0.0006, ν=0.30. Use relative strains, not absolute millimetres, and measure both directions at the same load state. Poisson’s ratio is a dimensionless measured material response in the small-strain elastic range. Identify axial and transverse directions before applying the minus sign. Anisotropic materials need directional elastic constants; one scalar ratio cannot describe every orientation or nonlinear response.

Can ν be treated as constant for every material?+

Only within an appropriate material model and range of conditions. Linear-elastic isotropic materials often use a constant ν. Anisotropy, temperature, microstructure or large nonlinear strain can change the response. A value measured in one direction should not be reused for a composite in another orientation without justification. Poisson’s ratio is a dimensionless measured material response in the small-strain elastic range. Identify axial and transverse directions before applying the minus sign. Anisotropic materials need directional elastic constants; one scalar ratio cannot describe every orientation or nonlinear response.

How do I check the result and units for Poisson’s ratio?+

Write the quantity and unit from the variable list next to every value: ν dimensionless; both strains dimensionless. Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example For εaxial=0.002 and εtransverse=−0.0006, ν=−(−0.0006)/0.002=0.30. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. Poisson’s ratio is a dimensionless measured material response in the small-strain elastic range. Identify axial and transverse directions before applying the minus sign. Anisotropic materials need directional elastic constants; one scalar ratio cannot describe every orientation or nonlinear response.

Which model limit matters for Poisson’s ratio?+

The background states this starting condition: Ordinary isotropic materials typically contract laterally in tension, which motivates the minus sign. The definition describes measured material. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. Poisson’s ratio is a dimensionless measured material response in the small-strain elastic range. Identify axial and transverse directions before applying the minus sign. Anisotropic materials need directional elastic constants; one scalar ratio cannot describe every orientation or nonlinear response.

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Which formula describes Poisson’s ratio?

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Subject references and further reading

This list names subject references and further reading. A page or section number appears when it is part of the reference; these are not verbatim quotations.

[1]Engineering LibreTexts (Roylance), Constitutive Relations, linear-elastic strain and Poisson ratio.

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Symbol

Meaning

Unit

ν

Poisson’s ratio in the stated material range

dimensionless

εquer

Strain perpendicular to loading

dimensionless

εaxial

Strain along the loading direction

dimensionless

Derivation & background

Exam blueprint

Validity range

Derivation steps

In a uniaxial test, Poisson’s ratio is the negative ratio of transverse strain to axial strain.

1State the model assumptions and list quantities with units.
2Substitute into ν = −εquer/εaxial; convert quantities to SI first.
3For εaxial=0.002 and εtransverse=−0.0006, ν=−(−0.0006)/0.002=0.30.

Frequently asked questions about Poisson’s ratio

What does the minus sign mean?+

How do I determine ν from measurements?+

Can ν be treated as constant for every material?+

How do I check the result and units for Poisson’s ratio?+

Which model limit matters for Poisson’s ratio?+

Retain Poisson’s ratio for exams

Create a curated FSRS exam set for ν = −εquer/εaxial: formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.

Curated formula set · LaTeX · FSRS spaced repetition · import from Essential