Poisson's Ratio (σ)

When a rod or wire is stretched by a tensile force along its length, careful measurement shows that it does not merely become longer -- it also becomes very slightly thinner, contracting a small amount in every direction perpendicular to the applied force (and, conversely, a rod compressed along its length bulges very slightly outward, perpendicular to the force). This sideways response to a lengthwise deformation is captured by Poisson's ratio, denoted $\sigma$ (sometimes written $\nu$), defined as the negative of the ratio of lateral strain to longitudinal strain:

If a rod of original length $L$ and original diameter $d$ is stretched so that its length increases by $\Delta L$ (longitudinal strain $= \Delta L/L$) and its diameter simultaneously decreases by $\Delta d$ (lateral strain $= -\Delta d/d$, negative because the diameter is decreasing), then

The minus sign in the original definition exists for exactly the same reason as the one in the bulk modulus's definition: a stretch (positive longitudinal strain) always produces a lateral contraction (negative lateral strain), so without the minus sign the ratio would always come out negative; with it, $\sigma$ comes out as a positive number for essentially every ordinary material, matching the sign convention already used for $Y$, $\eta$, and $K$.

Both lateral strain and longitudinal strain are dimensionless ratios (each measured as a fractional change in a length), so their ratio, Poisson's ratio $\sigma$, is itself a pure, dimensionless number -- unlike $Y$, $\eta$, and $K$, it carries no unit at all, and this is a useful way to recognise it immediately in a numerical problem. …