Lesson 7 of 9 · 6 min

Poisson's ratio and elastic energy

NCERT §8.5.4, §8.5.5

Meera notices that when a rubber band is stretched it also gets thinner. Does the steel tie-rod do the same when it stretches 1.59 mm? And where does the work of stretching it go?

The lesson in notes

In short

Stretching a wire also thins it. The strain at right angles to the applied force is the lateral strain, Δd/d for a wire of diameter d.

Within the elastic limit lateral strain is proportional to longitudinal strain (Simon Poisson). Poisson's ratio = (Δd/d)/(ΔL/L) = (Δd/ΔL)(L/d).

Poisson's ratio is a ratio of two strains, so it is a pure number with no unit or dimensions, and it depends only on the material: 0.28 to 0.30 for steels and about 0.33 for aluminium alloys.

Stretching a wire does work against the interatomic forces, and that work is stored as elastic potential energy.

For an extension l of a wire of length L and area A, the force is F = YA(l/L). Integrating F dl from 0 to l gives W = ½ YA l²/L.

So U = ½ × Y × strain² × volume = ½ × stress × strain × volume, and the energy per unit volume is u = ½ σε.