Poisson's Ratio - Compression

Let’s take a look back at tensile stress, from back in February. We said that the axial pulling of a isotropic bar creates axial or tensile stress. Some definitions:

• Axial – lengthwise along the bar
• Isotropic – often a metal, where the “crystal” structure in the material is uniform (wood and carbon fiber are not prismatic because of grain and stress biases, for example)
• Tensile stress – mechanical stress that builds up as a result of something being pulled

Whenever tensile stress is applied to something as a result of pulling, that something tends to elongate. This much is intuitive and obvious. What we don’t really pay attention to is what happens in the other direction, i.e. the lateral direction.

Take out your favorite rubber eraser (a not just a remnant nub of an eraser!). I am using a new Sanford Magic Rub for this demonstration. (Note that a rubber eraser is not entirely prismatic as we require, but it does the job very well.) Now try to evenly pull the two ends of the erase apart lengthwise, as hard as you can. As you can expect, you’ll notice that the eraser elongates slightly until you stop pulling, at which time it returns to its original length. What happens in the other direction; does the width change? And what happens if we compress the eraser? Does the opposite effect occur?

The concept we use when we consider this is called the Poisson effect. Poisson’s ratio ν (nu) is a measure of the Poisson effect.

From Wikipedia:

When a sample cube of a material is stretched in one direction, it tends to contract (or occasionally, expand) in the other two directions perpendicular to the direction of stretch. Conversely, when a sample of material is compressed in one direction, it tends to expand (or rarely, contract) in the other two directions.

Poisson’s ratio depends upon the specific material and can be determined experimentally. Its technical definition is the negated ratio of transverse strain to axial strain. Poisson’s ratio ranges from -1.0 to +0.5.

A positive Poisson’s ratio means that the material contracts in the transverse direction as it stretches in the axial direction. A negative ratio means that the material bulges in the transverse direction as it stretches in the axial direction. Similarly, if the material is compressed, it would bulge (positive ratio) or contract (negative ratio).

Most metals hover about a Poisson’s ratio if 0.33. Here are a few examples: Copper is 0.33, Gold is 0.42, and Stainless steel is 0.30. What about our rubber eraser?

Did you notice that as you pulled on it lengthwise, its midsection started to contract? And when you compressed it lengthwise (which might be difficult because it cannot be allowed to bend), you should see that the eraser gets fatter.

As it turns out rubber has a Poisson’s ratio of about 0.5, which makes it one of the best showcases of the Poisson effect. Does our demo agree with the +0.5? Sure it does!

There are a few caveats for the Poisson effect to occur though, but we’ll save that for another time. For most practical purposes though, this is applicable and is quite awesome. (For the engineers… It isn’t safe to use a material’s Poisson’s ratio when we exceed the plastic region of stress. When objects start to permanently deform, we have bigger problems, to say the least.)

Pressure Formula

Stress, it turns out, is exactly the same. It has the same units, and it conceptually and intuitively has the same behavior. They go hand in hand, so I’ll refer the pair interchangeably. While stress isn’t derived from engineering, it is a fundamental science tool that nearly all engineers use.

So, what is it exactly? In every day life, we deal with pressure in our car and bike tires. We have tire gauges to tell us the air pressure inside the tire, in “psi”. The metric equivalent is the Pascal (or Pa), where the force is in Newtons, and the area is in squared meters. I guess to follow the “psi” format and for purposes of demonstration only, it’ll supposedly be Npm, or Newton per square meter.

For now, we’ll simplify the whole deal by looking only at positive axial normal stresses. It’s positive because we’re stretching and not compressing. (It is also called tensile stress.) It’s axial and normal because the area we look at is perpendicular to the direction we’re pulling; for now, we’re not pulling at some angle to the cross-sectional area in the general formula.

In everyday engineering, we tend to deal instead with ksi (kilopounds per square inch) and MPa or GPa (megapascals or gigapascals, respectively). One psi or one Pa is far too small to deal with because a pound-force is too small to act over a square inch and a square meter is too large for a Newton to act upon. Don’t worry. This’ll make more sense when we discuss really strong bars and beams made out of steel and aluminum, as opposed to rulers made out of plastic.

The next time to go walking or driving over a bridge, for example, you can hopefully realize that all the metal cables and beams are working in tandem with huge amounts of tensile stress to support you, the other cars, and itself. The cabling of a suspension bridge, for example, is completely engineered with the fundamental principle of tensile stress of force divided by area.

That’s it for now.

But check out the video below. It shows what happens to a steel bar when there is too much tensile stress in the bar from too much pulling force.

If you notice at 0:27, the steel actually appears to stretch, a process called necking. At 0:30, there is too much force and stress to the point that the steel broke in two. In future posts, we’ll discuss these magical stretching and breaking points, and how we can calculate how much force it took to fracture this piece of steel.

Take care.