How this instrument works
Pull a rubber eraser lengthwise and it visibly narrows in the middle; press a cork into a bottleneck and it barely swells sideways at all. Poisson's ratio, ν, is the number that separates those two behaviors: the transverse strain a material develops divided by the axial strain that caused it, ν = −ε_transverse ⁄ ε_axial. Siméon Poisson worked out the relationship in 1811 while developing the theory of elasticity, and the ratio still carries his name in every mechanics-of-materials textbook written since.
The minus sign exists because, for almost every ordinary solid, stretching along one axis and thinning across it point in opposite directions — axial strain reads positive while transverse strain reads negative, so the raw division comes out negative and the formula flips it back. Isotropic solids under simple tension keep ν between roughly 0 and 0.5: cork sits near 0 because it hardly contracts sideways, structural steel clusters around 0.27 to 0.30, and rubber climbs toward 0.5, the value at which a stretched shape changes but its volume does not.
The relationship only holds in the small-strain, linear-elastic range — the same few tenths of a percent a bonded strain gauge measures before a metal specimen yields or a rock core fractures. Push past that point and the ratio drifts, which is why a materials lab bonds two gauges, one along the load and one across it, and reads both at once instead of trusting a single handbook figure. Geotechnical engineers pull the same number out of a triaxial soil test to size a foundation, and a small class of engineered lattices called auxetic structures break the usual pattern outright, growing wider instead of thinner as they stretch.
- Enter the Axial strain — the fractional length change measured along the direction of pull, positive for stretching and negative for compression.
- Enter the Transverse strain — the fractional width or diameter change measured across the load direction; for ordinary materials this carries the opposite sign to the axial value.
- Type both fields as plain decimals, not percentages — 0.003 for a 0.3 percent change — since strain is already a length ratio and needs no unit of its own.
- Read the Poisson's ratio result: a figure near 0.3 matches most metals, near 0 matches cork-like materials, and near 0.5 matches a nearly incompressible rubber.
Worked example — a steel tension coupon
A structural steel coupon sits in a tensile testing machine with two strain gauges bonded to its surface: one running along the pull, one wrapped around the gauge section. At the recorded load, the Axial strain field reads 0.003 — a 0.3 percent stretch — while the Transverse strain field reads −0.0009, a 0.09 percent narrowing across the section.
The formula gives ν = −(−0.0009 ⁄ 0.003) = 0.3 exactly. That figure lands squarely inside the 0.27-to-0.30 band published for structural steel in mechanics-of-materials references, which is precisely the cross-check a materials lab runs to confirm a specimen is behaving as expected before trusting the rest of its test data.
Questions
Why does the formula have a minus sign?
Because axial stretching and transverse shrinking normally carry opposite signs — axial strain positive, transverse strain negative — so their raw ratio comes out negative. The minus sign in ν = −ε_transverse ⁄ ε_axial flips that back to a positive figure for the vast majority of everyday materials, which is the convention every materials handbook expects.
What is a typical range for Poisson's ratio?
For ordinary isotropic solids under simple tension, ν runs from about 0 to 0.5. Cork sits near 0 since it barely bulges sideways, which is why it works as a bottle stopper. Most metals, including structural steel, cluster around 0.27 to 0.34. Rubber and other near-incompressible materials approach 0.5, the theoretical ceiling.
Can Poisson's ratio be greater than 0.5?
Not for an isotropic, linear-elastic material — 0.5 is the thermodynamic upper bound, reached when a material holds constant volume as it deforms, which is why it is called incompressible at that limit. Anisotropic materials, such as wood measured along certain grain directions, can show ratios above 0.5 or below 0 without breaking any physical law.
What does a negative Poisson's ratio mean?
It means the material gets wider, not narrower, when stretched — the opposite of nearly everything handled day to day. These auxetic materials are usually engineered on purpose: re-entrant honeycomb structures and certain open-cell foams are built so their internal geometry unfolds sideways under axial tension instead of pinching inward.
Does Poisson's ratio have units?
No. It is one strain divided by another, and strain itself is a length change divided by a length, so the units cancel top and bottom. Enter the Transverse strain and Axial strain fields as plain decimals — 0.003, not 0.3 percent or 3 mm — and the result comes out unitless too.
How is Poisson's ratio used beyond a single test?
It links the elastic constants engineers design with: shear modulus, bulk modulus, and Young's modulus are all related through ν, so one measured ratio lets a finite-element model reconstruct a material's full elastic behavior from a single tension test instead of three separate ones.