Stretch a bar and it thins. The ratio of that thinning to the stretch is one number, ν — dimensionless, and for any ordinary material it sits between −1 and ½. Down the center, data flows: the strains go in, the engine computes the width loss and the volume change, the verdict comes out. The blue team builds and defends it; the red team tries to break it.
source Poisson, S. D., Mémoire sur l'équilibre et le mouvement des corps élastiques homogènes, Mém. Acad. Sci. 8 (1829) — where the transverse-to-axial strain ratio first appears. No stable open scan; cited by author / title / year. Rendered, not quoted.
Pull a bar along x by axial strain ε. It contracts on the two transverse axes by the same amount, scaled:
εy = εz = −ν·ε → ν = −εlat / εaxial. Sum the three and, to first order, the fractional volume change is dV/V = ε(1 − 2ν).
Live, for the current material and stretch:
| quantity | value | sign |
|---|
An isotropic solid has only two independent elastic constants. Give it the-youngs-modulus E (the stiffness) and this ratio ν (the volume response), and everything else follows — the bulk modulus K = E / 3(1−2ν) and the shear modulus G = E / 2(1+ν) of the-shear.
Stretch thins the bar — Poisson's ratio, bounded by ½, is the missing half of the-hookes-law: Hooke says how hard it resists, Poisson says which way it moves while it does. Each sphere is the next one's premise.
The blue team's live check: re-run the physical law on a reference material and confirm ν is in bounds, the volume grows under tension, and K, G stay positive. If red tampers, this badge is where it shows.
Two numbers feed the engine: the axial strain ε you apply, and the material's ratio ν. Everything downstream — lateral contraction, volume change, moduli — is computed from just these two. Four real materials, and the sign their volume takes under tension:
| material | ν | dV/V under tension |
|---|---|---|
| cork | ~0.00 | grows most (≈ ε) |
| steel | ~0.30 | grows (0.4·ε) |
| rubber | ~0.49 | barely grows |
| auxetic foam | ~−0.30 | grows fast, widens |
ν = ½ is the wall: incompressible, zero volume change. Above it the volume would shrink under tension — impossible for an isotropic solid. That is what you feed the panel below.
Dashed = the bar at rest; solid = deformed. Every number is computed from ε and ν on the spot, never looked up.
What the machine produces, proven: ν = −εlat/εaxial is dimensionless and positive for a normal (thinning) bar; the volume law dV/V = ε(1−2ν) makes ν = ½ exactly incompressible and ν < ½ grow under tension; the isotropic bound is −1 < ν < ½, exactly where K and G stay positive. The current form's readout is above; these invariants are the output.
The blue team's witness (left) confirms these live; the red team (right) tries to make them wrong.
And ν is not a fixed fingerprint: it drifts with temperature, loading rate and pressure (polymers stiffen and their ν shifts near the glass transition). The engine reports the ideal linear-elastic, isotropic, small-strain limit — a choice, not the whole truth.
"Poisson's ratio is always positive." Cut. Auxetic materials (re-entrant foams, some crystals) have ν < 0 — pull them and they get wider. The engine admits down to −1.
"Poisson proved ν = ¼ for every material." Cut. His molecular ("rari-constant") model forced a single value of ¼; measurement killed the uni-constant theory. Real isotropic solids span the whole range.
"Rubber has ν = 0.5 exactly." Kept, corrected. Rubber is near-incompressible, ν ≈ 0.49–0.4999 — never exactly ½, since that would need infinite bulk modulus.
The red team's move: force ν = 0.8 on the witness's reference material — past the physical wall at ½. The blue team's witness (window 7) is watching.
At ν = 0.8 the volume law dV/V = ε(1−2ν) = −0.6ε goes negative under tension: the bar would shrink in volume while stretched, and the bulk modulus goes negative. The witness recomputes, disagrees, and turns red. Nothing is faked; the attack is real and it is caught.