Stretch One Way and It Shrinks the Other
Stretch a rubber band out long and you can see it thin in the middle as it lengthens. You only pulled in one direction, yet the perpendicular direction shrank on its own. This is not just a rubber thing. Metals, concrete, nearly every material does it, only by different amounts. The ratio of that sideways shrinking to the lengthwise stretch is the Poisson's ratio ν, one more constant fixed for each material. It looks like a small effect, but it reaches surprisingly deep: whether the volume changes, and how the E and G you met earlier are tied together in one relation. This is the last piece of the A track.
First, just see the effect. Pull the bar out long. As it lengthens, the width quietly narrows. The dashed line is the original shape, and the more you pull, the thinner it gets. It feels a little odd: you clearly applied no sideways force, so why does the side shrink? Look closely at the material and, as the atoms spread apart along the pulling direction to lengthen, they draw inward sideways. To grow longer one way, it borrows material from the other. Stretch and it narrows, this is the first impression of the Poisson effect.
Now attach a number to the effect. Call the lengthwise strain εaxial and the sideways strain εlateral. Since the side shrinks, its sign is opposite, negative. The Poisson's ratio is that ratio with a minus sign in front. ν = − εlateralεaxial. The minus is there because the lateral strain is already negative, and it makes ν come out as a clean positive number. Drag εaxial. Stretch lengthwise and the width shrinks by 0.3 times as much. The ratio stays constant. Pull harder and ν does not change, because ν is not the size of the strain but a property the material carries.
Here is a fun question. If it lengthens while the sides shrink, what happens to the volume? Do the two cancel and leave it unchanged? Work it out and the fractional volume change is about ΔVV ≈ ε(1 − 2ν). Drag ν. Most materials have ν below 0.5, so pulling makes the volume grow slightly: the sideways shrinking does not fully eat up the lengthwise stretch. But when ν hits exactly 0.5, the 1 − 2ν becomes zero and the volume does not change at all. This is incompressibility. Rubber is close to this, keeping nearly the same volume whether stretched or squeezed. It is also why ν cannot exceed 0.5: beyond it, pulling would strangely shrink the volume.
The value of ν differs by material. Switch with the buttons. Most metals have ν near 0.3. Steel, aluminum, copper are all about the same, so when a problem gives no value, taking 0.3 for a metal is the usual move. At the far end is rubber, with ν close to 0.5, so pulling shrinks the width a lot. It is nearly incompressible. And the fun one is cork, with ν near 0: pull it and the sides barely shrink. That is exactly why a cork pushes into a bottle so well, it does not bulge sideways as you press it in. So ν falls between 0 and 0.5.
Finally, the punch line for why ν matters. Earlier you met the tensile stiffness E and the shear stiffness G separately. But in an isotropic material the two are not independent: ν ties them together. G = E(2(1 + ν)). Drag ν. Even with E held fixed, ν sets G. For a metal with ν of 0.3, G is about 0.385 times E, so steel's E of 200 GPa gives a G of about 77 GPa. Thanks to this relation, knowing any two of the three gives the third for free. E, G, and ν are really one family with only two degrees of freedom. With this, the A track, the foundations of stress and strain, comes to a close.