what is Poisson's ratio
Pull a rubber band long and you see it lengthen while the middle grows thin. You pulled in only one direction, yet the direction perpendicular to it shrinks on its own. This is not only rubber. Metals and concrete, nearly every material, do this, and only the amount differs. The ratio of the sideways shrink to the lengthwise growth is called Poisson's ratio. It is one more constant fixed for each material. The effect looks small, but it reaches surprisingly deep, into whether the volume changes and into how the E and G met earlier are tied by one formula. It is the last piece of thread A. The sideways shrink is part of the same pull.
Look first at the effect itself. Pull a bar long and, as it lengthens, the sideways width quietly narrows. The dashed outline is the original shape, and the harder you pull the thinner it gets. Intuition finds this odd. You gave no force sideways at all, so why does the side shrink? Look finely at the material and the atoms spread along the axis in order to lengthen, while sideways they are drawn toward each other. To grow long on one side, it borrows flesh from the other. Pull and it narrows. That is the first impression of the Poisson effect.
Now put a number on the effect. Set the lengthwise strain and the sideways strain side by side. The side shrinks, so its sign is the opposite, a negative. Poisson's ratio is the ratio of the two with a minus sign attached. ν = − εlateralεaxial. The minus sign is there because the sideways strain is negative to begin with, and the sign makes the ratio a clean positive. Move the lengthwise strain and, as it grows, the side shrinks by 0.3 times that much. The ratio is always steady. Pull harder and the ratio does not change. The ratio is not the size of the deformation. It is a property the material has.
An interesting question follows. If it lengthens while the side shrinks, what happens to the volume? Do the two cancel and leave it as it was? Work it out and the rate of volume change is about ΔVV ≈ ε(1 − 2ν). Move the ratio. Ordinary materials have a ratio smaller than 0.5, so pulling makes the volume grow a little. The sideways shrink cannot eat the whole lengthwise growth. But when the ratio is exactly 0.5, 1 − 2ν becomes 0 and the volume does not change at all. That is incompressibility. Rubber sits near there, so whether you stretch it or squeeze it the volume stays almost the same. The ratio cannot pass 0.5 for the same reason. Then pulling would make the volume shrink, which is a strange thing.
The value differs by material. Change it with the buttons. Most metals have a ratio near 0.3. Steel, aluminum, and copper are all similar. So when a problem does not give the value, taking 0.3 for a metal is the usual move. At the other end is rubber. Its ratio is near 0.5, so pulling shrinks the side a lot and it is nearly incompressible. And cork is the interesting one. Its ratio is near 0, so pulling hardly shrinks the side at all. That is why a cork stopper goes into a bottle without bulging sideways. The ratio therefore sits between 0 and 0.5.
Last, one blow that shows why the ratio matters. Earlier the tensile stiffness E and the shear stiffness G were met separately. In an isotropic material they are not independent. Poisson's ratio ties the two. G = E(2(1 + ν)). Move the ratio and, even with E held fixed, the ratio sets G. If the ratio is 0.3, as in a metal, G is about 0.385 times E, so for steel's E of 200 GPa the G is about 77 GPa. Thanks to this formula, knowing two of the three brings the third out on its own. E, G, and Poisson's ratio, these three constants are in fact one family with only two degrees of freedom. With this, thread A, the foundation of stress and deformation, ends.