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Material fundamentals · Coupling

Poisson’s ratio and coupled deformation

When wood stretches along the grain, what happens across it?

Poisson’s ratio describes transverse strain caused by axial loading. In wood, the direction of loading and the direction of observation must both be specified.

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Start with something familiar

Gently stretching an eraser often makes it a little narrower sideways. That lateral change is not an extra load; it is coupled to the stretching. Poisson’s ratio puts a number on that coupling.

Original shape

Axial extension with lateral contraction

Under load

Axial extension with lateral contraction

Diagram · An axially stretched specimen becomes narrower in this positive-Poisson-ratio example. Both changes are exaggerated; the numbers use the actual strain.

  1. Notice what changes

    First measure relative length change, not only millimetres. Stretching a 100 mm strip by 1 mm is 1% axial strain.

  2. Understand why

    Now compare the relative width change. With ν = 0.3, an imposed 1% axial stretch gives 0.3% lateral contraction in the simple uniaxial example.

  3. Connect it to wood and making

    For wood, label both directions. Loading along L and observing R is a different experiment from loading along R and observing L. The two values are connected, not generally equal.

Put names to the ideas

01

A strain ratio

Define ν_LR here as minus radial strain divided by longitudinal strain under longitudinal uniaxial loading. The minus sign makes ordinary lateral contraction positive.

02

Subscript order matters

Different references may use different index conventions. With this convention, reciprocal loading gives ν_LR/E_L = ν_RL/E_R, not generally ν_LR = ν_RL.

03

Part of the full elastic model

The transverse coupling terms help specify a plate’s elastic response. Orthotropic symmetry leaves three independent Poisson ratios alongside three Young’s and three shear moduli.

νLR=−εRεL,νLREL=νRLER\nu_{LR}=-\frac{\varepsilon_R}{\varepsilon_L},\qquad \frac{\nu_{LR}}{E_L}=\frac{\nu_{RL}}{E_R}

The minus sign in ν = −εtransverse/εaxial makes an ordinary contraction give a positive result. The reciprocity equation connects paired directions through their Young’s moduli.

A worked example

If axial strain is 0.001 and radial strain is −0.0003, this convention gives ν_LR = 0.3. This is an arithmetic example, not a recommended constant for every spruce blank.

Explore it yourself

Check the reference’s index definition before importing Poisson ratios into a model. Use a physically consistent set of orthotropic constants.

CONTROLLED TEACHING MODEL
Axial strain
1.00 %
Transverse strain
-0.30 %
ε transverseε axial
A positive-Poisson-ratio uniaxial example, with imposed strain rather than imposed force. The shown 0–0.5 slider is an example range, not a universal anisotropic bound.

Check your understanding

Is a 0.3% width contraction always 0.3 mm?

References

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